Combined measurements of velocity components and temperature in a turbulent Rayleigh–Bénard convection flow at a low Prandtl number of $Pr=0.029$ and Rayleigh numbers of $10^{6}\leqslant Ra\leqslant 6\times 10^{7}$ are conducted in a series of experiments with durations of more than a thousand free-fall time units. Multiple crossing ultrasound beam lines and an array of thermocouples at mid-height allow for a detailed analysis and characterization of the complex three-dimensional dynamics of the single large-scale circulation roll in a cylindrical convection cell of unit aspect ratio which is filled with the liquid metal alloy GaInSn. We measure the internal temporal correlations of the complex large-scale flow and distinguish between short-term oscillations associated with a sloshing motion in the mid-plane as well as varying orientation angles of the velocity close to the top/bottom plates and the slow azimuthal drift of the mean orientation of the roll as a whole that proceeds on a time scale up to a hundred times slower. The coherent large-scale circulation drives a vigorous turbulence in the whole cell that is quantified by direct Reynolds number measurements at different locations in the cell. The velocity increment statistics in the bulk of the cell displays characteristic properties of intermittent small-scale fluid turbulence. We also show that the impact of the symmetry-breaking large-scale flow persists to small-scale velocity fluctuations thus preventing the establishment of fully isotropic turbulence in the cell centre. Reynolds number amplitudes depend sensitively on beam-line position in the cell such that different definitions have to be compared. The global momentum and heat transfer scalings with Rayleigh number are found to agree with those of direct numerical simulations and other laboratory experiments.

The low-dimensional dynamics of the structures in a turbulent wall flow are studied by means of numerical simulations. These are made both ‘minimal’, in the sense that they contain a single copy of each relevant structure, and ‘autonomous’ in the sense that there is no outer turbulent flow with which they can interact. The interaction is prevented by a numerical mask that damps the flow above a given wall distance, and the flow behaviour is studied as a function of the mask height. The simplest case found is a streamwise wave that propagates without change. It takes the form of a single wavy low-velocity streak flanked by two counter-rotating staggered quasi-streamwise vortices, and is found when the height of the numerical masking function is less than δ+1 ≈ 50. As the mask height is increased, this solution bifurcates into an almost-perfect limit cycle, a two-frequency torus, weak chaos, and full-edged bursting turbulence. The transition is essentially complete when δ+1 ≈ 70, even if the wall-parallel dimensions of the computational box are small enough for bursting turbulence to be metastable, lasting only for a few bursting cycles. Similar low-dimensional dynamics are found in somewhat larger boxes, containing two copies of the basic structures, in which the bursting turbulence is self-sustaining.

A geostrophic, hydrostatic, frontal or filamentary flow adjusts conservatively to accommodate a surface gravity wave field with wave-averaged, Stokes-drift vortex and Coriolis forces in an altered balanced state. In this altered state, the wave-balanced perturbations have an opposite cross-front symmetry to the original geostrophic state; e.g. the along-front flow perturbation is odd-symmetric about the frontal centre while the geostrophic flow is even-symmetric. The adjustment tends to make the flow scale closer to the deformation radius, and it induces a cross-front shape displacement in the opposite direction to the overturning effects of wave-aligned down-front and up-front winds. The ageostrophic, non-hydrostatic, adjusted flow may differ from the initial flow substantially, with velocity and buoyancy perturbations that extend over a larger and deeper region than the initial front and Stokes drift. The largest effect occurs for fronts that are wider than the mixed layer deformation radius and that fill about two-thirds of a well-mixed surface layer, with the Stokes drift spanning only the shallowest part of the mixed layer. For even deeper mixed layers, and especially for thinner or absent mixed layers, the wave-balanced adjustments are not as large.

The zero-pressure-gradient turbulent boundary layer over a flat plate was perturbed by a temporally oscillating, spatial impulse of roughness, and the downstream response of the flow field was interrogated by hot-wire anemometry and particle-image velocimetry. The key features common to impulsively perturbed boundary layers, as identified in Jacobi & McKeon (J. Fluid Mech., 2011), were investigated, and the unique contributions of the dynamic perturbation were isolated by contrast with an appropriately matched static impulse of roughness. In addition, the dynamic perturbation was decomposed into separable large-scale and small-scale structural effects, which in turn were associated with the organized wave and roughness impulse aspects of the perturbation. A phase-locked velocity decomposition of the entire downstream flow field revealed strongly coherent modes of fluctuating velocity, with distinct mode shapes for the streamwise and wall-normal velocity components. Following the analysis of McKeon & Sharma (J. Fluid Mech., vol. 658, 2010, pp. 336–382), the roughness perturbation was treated as a forcing of the Navier–Stokes equation and a linearized analysis employing a modified Orr–Sommerfeld operator was performed. The experimentally ascertained wavespeed of the input disturbance was used to solve for the most amplified singular mode of the Orr–Sommerfeld resolvent. These calculated modes were then compared with the streamwise and wall-normal velocity fluctuations. The discrepancies between the calculated Orr–Sommerfeld resolvent modes and those experimentally observed by phase-locked averaging of the velocity field were postulated to result from the violation of the parallel flow assumption of Orr–Sommerfeld analysis, as well as certain non-equilibrium effects of the roughness. Additionally, some difficulties previously observed using a quasi-laminar eigenmode analysis were also observed under the resolvent approach; however, the resolvent analysis was shown to provide reasonably accurate predictions of velocity fluctuations for the forced Orr–Sommerfeld problem over a portion of the boundary layer, with potential applications to designing efficient flow control strategies. The combined experimental and analytical effort provides a new opportunity to examine the non-equilibrium and forcing effects in a dynamically perturbed flow.

Rotationally symmetric annular combustors are of practical importance because they generically resemble combustion chambers in gas turbines, in which thermoacoustically driven oscillations are a major concern. We focus on azimuthal thermoacoustic oscillations and model the fluctuating heat release rate as being dependent only on the local pressure in the combustion chamber. We study the dynamics of the annular combustor with a finite number of compact flames equispaced around the annulus, and characterize the flames’ response with a describing function. We discuss the existence, amplitude and the stability of standing and spinning waves, as a function of: (i) the number of the burners; (ii) the acoustic damping in the chamber; (iii) the flame response. We present the implications for industrial applications and the future direction of investigations. We then present as an example the first theoretical study of thermoacoustic triggering in annular combustors, which shows that rotationally symmetric annular chambers that are thermoacoustically unstable do not experience only stable spinning solutions, but can also experience stable standing solutions. We finally test the theory on one experiment with good agreement.

When fluid is withdrawn from a body of stratified fluid the surfaces of constant density are deformed towards the region of withdrawal. The equations describing the flow caused by withdrawal through a point sink in a two-layer unbounded system in which viscous forces dominate are formulated using the boundary-integral representation of Stokes flow. It is shown by dimensional and analytic arguments that surface tension between the layers is a necessary condition for the stability of an interfacial equilibrium in which only one fluid is withdrawn. The critical flow rate above which both fluids are withdrawn is determined numerically as a function of the capillary number. When the flow is supercritical a small adaptation of the numerical scheme allows the proportion of fluid withdrawn from each layer to be found. The various analyses and conclusions further our understanding of the physical processes that determine the compositional output of volcanic eruptions that tap an underlying stratified reservoir of magma.

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.

The instabilities of Saffman–Taylor viscous fingers are revisited experimentally in the standard linear channel as well as in wedges of angle θ0. The local destabilization of a finger occurs by a splitting of its tip and results in the formation of two branches separated by a fjord. It is shown that, in a first approximation, the central line of a fjord follows a curve normal to the successive profiles of stable fingers. These normal curves are computed analytically for the Saffman–Taylor finger in a linear cell and numerically for the wedges. The length of a fjord is critically sensitive to the position of the initial destabilization of the finger. The nearer to the tip it occurs, the longer the fjord will be. Assuming a uniform spatial distribution of the disturbances in the central part of the finger front it is possible to predict the size distribution of the lateral branches. In the linear channel the probability of branches larger than the channel width is negligible. For wedges of increasing angle the probability of large secondary branches increases. Finally, for wedges with θ0 larger than approximately 90° infinite fjords separating two long-lived structures are observed. Our experimental results also suggest a generalization of the definition of virtual cells. With this new definition it is possible to show that the increasing complexity of the patterns corresponds to a hierarchy of virtual cells of various sizes.

We consider the steady propagation of an air finger into a buckled elastic tube initially filled with viscous fluid. This study is motivated by the physiological problem of pulmonary airway reopening. The system is modelled using geometrically nonlinear Kirchhoff–Love shell theory coupled to the free-surface Stokes equations. The resulting three-dimensional fluid–structure-interaction problem is solved numerically by a fully coupled finite element method.

The system is governed by three dimensionless parameters: (i) the capillary number, Ca=μU/σ*, represents the ratio of viscous to surface-tension forces, where μ is the fluid viscosity, U is the finger's propagation speed and σ* is the surface tension at the air–liquid interface; (ii) σ=σ*/(RK) represents the ratio of surface tension to elastic forces, where R is the undeformed radius of the tube and K its bending modulus; and (iii) A∞=A*∞/(4R2), characterizes the initial degree of tube collapse, where A*∞ is the cross-sectional area of the tube far ahead of the bubble.

The generic behaviour of the system is found to be very similar to that observed in previous two-dimensional models (Gaver et al. 1996; Heil 2000). In particular, we find a two-branch behaviour in the relationship between dimensionless propagation speed, Ca, and dimensionless bubble pressure, p*b/(σ*/R). At low Ca, a decrease in p*b is required to increase the propagation speed. We present a simple model that explains this behaviour and why it occurs in both two and three dimensions. At high Ca, p*b increases monotonically with propagation speed and p*b/(σ*/R) ∝ Ca for sufficiently large values of σ and Ca. In a frame of reference moving with the finger velocity, an open vortex develops ahead of the bubble tip at low Ca, but as Ca increases, the flow topology changes and the vortex disappears.

An increase in dimensional surface tension, σ*, causes an increase in the bubble pressure required to drive the air finger at a given speed; p*b also increases with A*∞ and higher bubble pressures are required to open less strongly buckled tubes. This unexpected finding could have important physiological ramifications. If σ* is sufficiently small, steady airway reopening can occur when the bubble pressure is lower than the external (pleural) pressure, in which case the airway remains buckled (non-axisymmetric) after the passage of the air finger. Furthermore, we find that the maximum wall shear stresses exerted on the airways during reopening may be large enough to damage the lung tissue.

A simple numerical scheme for the calculation of the motion of shock waves in gases based on Whitham's theory of geometrical shock dynamics is developed. This scheme is used to study the propagation of shock waves along walls and in channels and the self-focusing of initially curved shockfronts. The numerical results are compared with exact and numerical solutions of the geometrical-shock-dynamics equations and with recent experimental investigations.

This paper describes an experimental study of the electromagnetic stirring in a mercury induction furnace. The 200 mm-diameter furnace is supplied with a single-phase electric current of frequency 50–4700 Hz. The flow pattern is measured by means of a special two-wire probe, which tracks the thermal wake behind a hot-film probe. The magnitudes of fluctuating velocities are measured by hot-film anemometry. Attention is focused on the influence on the mean and turbulent motion of the electromagnetic-skin depth, which is determined by the supply frequency. The measurements of the mean motion show that, for a fixed magnetic field, stirring is maximum when the value of the skin depth normalized by the pool radius is about 0.2, in agreement with previous theoretical predictions. Two turbulence regimes may be distinguished for different frequency ranges. At low frequency the various properties of the turbulence, such as the mean-square fluctuations, the integral scales and the turbulent dissipation rate, are almost uniform over the whole bath. However, at high frequency the turbulence is non-uniform; there is an increase in the turbulent fluctuations and dissipation rate and a decrease of the integral scale within the electromagnetic-skin depth near the wall.

The velocity fields in and around a deformed drop suspended in an arbitrary (albeit Stokesian) unbounded flow field are solved. The usefulness of the solution is demonstrated by solving the drag force and lateral migration of a drop suspended in an unbounded Poiseuillian field.

It is demonstrated that, due to the deformation of the drop, there exists a radial component of the settling velocity. The direction of the radial migration depends primarily on the product UHR (the Hadamard-Rybczynski terminal settling velocity) by U0 (the maximum Poiseuillian velocity). A positive product results in a lateral migration away from the location of maximum velocity; the converse also holds.

The prototype spin-up problem between infinite flat plates treated by Greenspan & Howard (1963) is extended to include the presence of an imposed axial magnetic field. The fluid is homogeneous, viscous, and electrically conducting. The resulting boundary initial-value problem is solved to first order in Rossby number by Laplace transform techniques. In spite of the linearization the complete hydromagnetic interaction is preserved: currents affect the flow and the flow simultaneously distorts the field. In part 1, we analyze the impulsively started time dependent approach to a final steady Ekman–Hartmann boundary layer on a single insulating flat plate. The transient is found to consist of two diffusively growing boundary layers, inertial oscillations, and a weak Alfvén wave front. In part 2, these one plate results are utilized in discussing spin-up between two infinite flat insulating plates. Two distinct and important hydromagnetic spin-up mechanisms are elucidated. In all cases, the spin-up time is found to be shorter than in the corresponding non-magnetic problem.

Flow in a partly liquid-filled, rotating, horizontal cylinder is analysed by means of finite-element numerical simulation. Of alternative methods for locating the free surface, a boundary collocation scheme with Newton-Raphson iteration converges. This method forces the residual in the normal-stress boundary condition to zero at a finite set of points on the liquid meniscus. Solutions of the steady, two-dimensional, incompressible flow problem show circumferential variation of the liquid-film thickness and corresponding pressure and velocity fields, including recirculation zones. The complications of an unknown meniscus location and a nonlinear normal-stress condition when surface tension is significant are illustrated. The finite-element method proves an effective and convenient tool for such flows, in which inertial, gravitational, pressure, viscous and capillary forces are all important.

A set of constitutive equations is derived to describe the time-dependent flow of a dilute suspension of identical rigid particles of arbitrary shape which are influenced by Brownian couples. The form of the orientation probability distribution for small departures from isotropy is found. The case of weak flows with strong Brownian effects is studied in detail, and the viscoelastic approximation and second-order-fluid limit of the constitutive equation are derived for a general particle shape. A full numerical solution is given for ellipsoids. The general nearly spherical particle is also considered and constitutive equations for general flow strengths are obtained for this case.

The flow of compositional gravity currents past circular cylinders mounted above a wall is investigated numerically. Two- and three-dimensional Navier–Stokes simulations are employed to quantify the force load on the cylinder, along with the friction velocity at the bottom wall near the cylinder, for Reynolds numbers in the range of 2000–45 000. While two-dimensional simulations accurately capture the impact stage, they are seen to overpredict the force and friction velocity fluctuations throughout the transient stage. Comparisons between gravity current and constant-density flows past circular cylinders show that the impact and transient stages are unique to gravity current flows. During the quasi-steady stage, on the other hand, the wake structures and the values of the drag, the peak-to-peak lift, the vortex shedding frequency and the friction velocity below the cylinder are comparable.

The friction velocity below the cylinder depends chiefly on the Reynolds number formed with the front velocity and the gap width. The maximum friction velocity at impact is about 60% larger than during the quasi-steady stage or in a constant-density flow. This raises the possibility of aggressive erosion behaviour at impact, which may occur in a spanwise localized fashion because of the larger friction velocity near the lobes.

In classical literature, blowout is described as loss of static stability of the combustion system whereas thermoacoustic instability is seen as loss of dynamic stability of the system. At blowout, the system transitions from a stable reacting state to a non-reacting state, indicating loss of static stability of the reaction. However, this simple description of stability margin is inadequate since recent studies have shown that combustors exhibit complex nonlinear behaviour prior to blowout. Recently, it was shown that combustion noise that characterizes the regime of stable operation is itself dynamically complex and exhibits multifractal characteristics. Researchers have already described the transition from combustion noise to combustion instability as a loss of multifractality. In this work, we provide a multifractal description for lean blowout in combustors with turbulent flow and thus introduce a unified framework within which both thermoacoustic instability and blowout can be described. Further, we introduce a method for predicting blowout based on the multifractal description of blowout.

We report on new patterns in high-speed flows of granular materials obtained by means of extensive numerical simulations. These patterns emerge from the destabilization of unidirectional flows upon increase of mass holdup and inclination angle, and are characterized by complex internal structures, including secondary flows, heterogeneous particle volume fraction, symmetry breaking and dynamically maintained order. In particular, we evidenced steady and fully developed ‘supported’ flows, which consist of a dense core surrounded by a highly energetic granular gas. Interestingly, despite their overall diversity, these regimes are shown to obey a scaling law for the mass flow rate as a function of the mass holdup. This unique set of three-dimensional flow regimes raises new challenges for extending the scope of current granular rheological models.