A review of analyses based upon anti-parallel vortex structures suggests that structurally stable dipoles with eroding circulation may offer a path to the study of vorticity growth in solutions of Euler’s equations in $\mathbb{R}^{3}$. We examine here the possible formation of such a structure in axisymmetric flow without swirl, leading to maximal growth of vorticity as $t^{4/3}$. Our study suggests that the optimizing flow giving the $t^{4/3}$ growth mimics an exact solution of Euler’s equations representing an eroding toroidal vortex dipole which locally conserves kinetic energy. The dipole cross-section is a perturbation of the classical Sadovskii dipole having piecewise constant vorticity, which breaks the symmetry of closed streamlines. The structure of this perturbed Sadovskii dipole is analysed asymptotically at large times, and its predicted properties are verified numerically. We also show numerically that if mirror symmetry of the dipole is not imposed but axial symmetry maintained, an instability leads to breakup into smaller vortical structures.

We present experimental studies of higher-order modes of the flow in turbulent thermal convection in cells of aspect ratio ($\unicode[STIX]{x1D6E4}$) 1 and 0.5. The working fluid is water with the Prandtl number ($Pr$) kept at around 5.0. The Rayleigh number ($Ra$) ranges from $9\times 10^{8}$ to $6\times 10^{9}$ for $\unicode[STIX]{x1D6E4}=1$ and from $1.6\times 10^{10}$ to $7.2\times 10^{10}$ for $\unicode[STIX]{x1D6E4}=0.5$. We found that in $\unicode[STIX]{x1D6E4}=1$ cells, the first mode, which corresponds to the large-scale circulation (LSC), dominates the flow. The second mode (quadrupole mode), the third mode (sextupole mode) and the fourth mode (octupole mode) are very weak, on average these higher-order modes each contains less than 4 % of the total flow energy. In $\unicode[STIX]{x1D6E4}=0.5$ cells, the first mode is still the strongest but less dominant, the second mode becomes stronger which contains 13.7 % of the total flow energy and the third and the fourth modes are also stronger (containing 6.5 % and 1.1 % of the total flow energy respectively). It is found that during a reversal/cessation, the amplitude of the second mode and the remaining modes experiences a rapid increase followed by a decrease, which is opposite to the behaviour of the amplitude of the first mode – it decreases to almost zero then rebounds. In addition, it is found that during the cessation (reversal) of the LSC, the second mode dominates, containing 51.3 % (50.1 %) of the total flow energy, which reveals that the commonly called cessation event is not the cessation of the entire flow but only the cessation of the first mode (LSC). The experiment reveals that the second mode and the remaining higher-order modes play important roles in the dynamical process of the reversal/cessation of the LSC. We also show direct evidence that the first mode is more efficient for heat transfer. Furthermore, our study reveals that, during the cessation/reversal of the LSC, $Nu$ drops to its local minimum and the minimum of $Nu$ is ahead of the minimum of the amplitude of the LSC; and reversals can be distinguished from cessations in terms of global heat transport. A direct velocity measurement reveals the flow structure of the first- and higher-order modes.

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.

We develop a set of equations to explore the behaviour of cooling elastic-plated gravity currents for constant influx conditions. In particular, we introduce a temperature-dependent viscosity to couple the flow thermal structure with the velocity field. We show that this coupling results in important deviations from the isoviscous case. In particular, the bending and gravity asymptotic regimes, characteristic of the isoviscous case, both split into three different thermal phases: a first ‘hot’ isoviscous phase, a second phase where the spreading rate drastically decreases and the flow thickens and a third ‘cold’ isoviscous phase. The viscosity that controls the spreading rate differs in both asymptotic regimes; it is the average viscosity of a small peeling region at the current tip in the bending regime and the average flow viscosity in the gravity regime. In both regimes, we characterize the evolution of the thermal anomaly and determine the time scale of the phase changes in terms of the Péclet number and of the viscosity contrast. Finally, we show that the evolution with bending and gravity can result in six different evolution scenarios depending on the combination of dimensionless numbers considered. We provide a phase diagram which summarizes them as a function of the flow Péclet number and viscosity contrast.

We report on a refined macroscopic model for slightly compressible gas slip flow in porous media developed by upscaling the pore-scale boundary value problem. The macroscopic model is validated by comparisons with an analytic solution on a two-dimensional (2-D) ordered model structure and with direct numerical simulations on random microscale structures. The symmetry properties of the apparent slip-corrected permeability tensor in the macroscale momentum equation are analysed. Slip correction at the macroscopic scale is more accurately described if an expansion in the Knudsen number, beyond the first order considered so far, is employed at the closure level. Corrective terms beyond the first order are a signature of the curvature of solid–fluid interfaces at the pore scale that is incompletely captured by the classical first-order correction at the macroscale. With this expansion, the apparent slip-corrected permeability is shown to be the sum of the classical intrinsic permeability tensor and tensorial slip corrections at the successive orders of the Knudsen number. All the tensorial effective coefficients can be determined from intrinsic and coupled but easy-to-solve closure problems. It is further shown that the complete form of the slip boundary condition at the microscale must be considered and an important general feature of this slip condition at the different orders in the Knudsen number is highlighted. It justifies the importance of slip-flow correction terms beyond the first order in the Knudsen number in the macroscopic model and sheds more light on the physics of slip flow in the general case, especially for large porosity values. Nevertheless, this new nonlinear dependence of the apparent permeability with the Knudsen number should be further verified experimentally.

In this study we examine two-component shear flows that are stable with respect to Kelvin–Helmholtz and to double-diffusive instabilities individually. Our focus is on diffusively stratified ocean regions, where relatively warm and salty water masses are located below cool fresh ones. It is shown that such systems may be destabilized by the interplay between shear and thermohaline effects, caused by unequal molecular diffusivities of density components. Linear stability analysis suggests that parallel two-component flows can be unstable for Richardson numbers exceeding the critical value for non-dissipative systems $(Ri=1/4)$ by up to four orders of magnitude. Direct numerical simulations indicate that these instabilities transform the initially linear density stratification into a series of well-defined horizontal layers. It is hypothesized that the combined thermohaline–shear instabilities could be ultimately responsible for the widespread occurrence of thermohaline staircases in diffusively stable regions of the World Ocean.

To reveal the scale dependence of the transport of turbulent kinetic energy in a channel flow, the constituents of a spectral energy budget equation are evaluated using direct numerical simulations. At each height in the buffer and overlap layers, the upward turbulent transport provides energy to the fluctuations at small scales, but removes it from those at large scales. Energy removed from the large scales in the overlap layer is carried upward to the centre of the channel and also downward to the vicinity of the wall. The downward energy fluxes at the large scales result in the well-known anomaly of turbulence intensity and the constituents of the budget equation near the wall. In the overlap layer the cospectrum of the spatial turbulent transport is scaled well by the mixing length. It shows that the structure of fluctuations involved in turbulent transport is self-similar in this layer, supporting the classical assumption. The cospectra of pressure–strain correlations are also evaluated. They are not scaled by the wall unit near the wall, but no symptom of the influence of large-scale structures is observed in the cospectra, at least for the present range of Reynolds numbers. Above the buffer layer the cospectra of the pressure–strain correlations are almost isotropic, and their relevant length scale is given by the mixing length in the overlap layer. The pressure–strain correlations are therefore rather local quantities.

The linear stability analysis predicts that the Mack second mode propagating in the boundary layer on a sufficiently cold plate can radiate acoustic waves into the outer inviscid flow. This effect, which is called as a spontaneous radiation (or emission) of sound, is associated with synchronization of the second mode with slow acoustic waves of the continuous spectrum. The theoretical predictions are confirmed by direct numerical simulations of wave trains and wave packets propagating in the boundary layer on a flat plate at free-stream Mach number 6 and wall-to-edge temperature ratio $T_{w}/T_{e}=0.5$. A non-uniform distribution of the wave packet components and the interference between the radiated acoustic waves result in an intricate pattern of the outer acoustic field. The spontaneous radiation of sound, in turn, strongly affects the wave packet in the boundary layer causing its elongation and modulation. This phenomenon may alter the downstream development of instability and delay the transition onset.

The rarefied gas flow phenomenon of thermophoresis is studied experimentally on a macroscopic spherical particle with a diameter of 5.1 cm for pressures ranging from 0.01 to 10 Pa (Knudsen numbers $Kn$ from 10 to 0.01, respectively). Size scaling with matching Knudsen numbers makes the results applicable to microscale particles such as aerosol droplets at atmospheric pressure. Two sets of measurements are presented. The first set, complemented by numerical modelling based on the solution of the ellipsoidal statistical Bhatnagar–Gross–Krook kinetic equation, is focused on a spherical particle of high thermal conductivity in close proximity to a heated wall. The second set is conducted for the same particle placed in a linear thermal gradient established between two parallel walls. Results show the first reproducible measurements of negative thermophoretic force acting on a spherical particle in the direction from cold to hot, with values of the order of 5 % of the maximum hot to cold force production.

We study non-inertial flows of single-phase yield stress fluids along uneven/rough-walled channels, e.g. approximating a fracture, with two main objectives. First, we re-examine the usual approaches to providing a (nonlinear) Darcy-type flow law and show that significant errors arise due to self-selection of the flowing region/fouling of the walls. This is a new type of non-Darcy effect not previously explored in depth. Second, we study the details of flow as the limiting pressure gradient is approached, deriving approximate expressions for the limiting pressure gradient valid over a range of different geometries. Our approach is computational, solving the two-dimensional Stokes problem along the fracture, then upscaling. The computations also reveal interesting features of the flow for more complex fracture geometries, providing hints about how to extend Darcy-type approaches effectively.

The disturbance evolution in a Mach-4.8 zero-pressure-gradient flat-plate boundary-layer flow altered by discrete three-dimensional roughness elements is investigated including a laminar breakdown scenario. Direct numerical simulation (DNS), as well as the biglobal linear stability theory based on two-dimensional eigenfunctions in flow cross-sections, are applied. Roughness elements with high ratios of spanwise width to streamwise length are compared at varying height and skewing angles with respect to the oncoming flow. For an oblique roughness, the element’s height is varied between 27 % and 68 % of the undisturbed boundary-layer thickness. Compared to a symmetric roughness element an obliquely placed element generates a more pronounced low-speed streak in the roughness wake. The linear stability analysis reveals the occurrence of eigenmodes that can be associated with the first and second modes in the flat-plate flow. At identical roughness height, larger amplification is found for the eigenmodes of the oblique set-up. The results are confirmed by unsteady DNS showing very good agreement with stability theory; transient-growth behaviour in the near wake of the roughness is of minor importance. The comparison of the results gained for adiabatic wind-tunnel flow conditions with those for atmospheric-flight conditions with wall cooling reveals significant differences in the wake vortex system with subsequent impact on the stability properties of the flow. The hot-flow cases are less unstable at identical roughness Reynolds numbers. A variation of the wall cooling shows that the roughness-wake first- and second-mode behaviour is similar to that of the flat-plate flow: wall cooling stabilizes the first-mode and destabilizes the second-mode instabilities of the roughness wake.

A numerical investigation of two locally applied drag-reducing control schemes is carried out in the configuration of a spatially developing turbulent boundary layer (TBL). One control is designed to damp near-wall turbulence and the other induces constant mass flux in the wall-normal direction. Both control schemes yield similar local drag reduction rates within the control region. However, the flow development downstream of the control significantly differs: persistent drag reduction is found for the uniform blowing case, whereas drag increase is found for the turbulence damping case. In order to account for this difference, the formulation of a global drag reduction rate is suggested. It represents the reduction of the streamwise force exerted by the fluid on a plate of finite length. Furthermore, it is shown that the far-downstream development of the TBL after the control region can be described by a single quantity, namely a streamwise shift of the uncontrolled boundary layer, i.e. a changed virtual origin. Based on this result, a simple model is developed that allows the local drag reduction rate to be related to the global one without the need to conduct expensive simulations or measurements far downstream of the control region.

The nonlinear dynamics of waves generated by the deformational oscillatory Marangoni instability in a two-layer film under the action of a spatial temperature modulation on the solid substrate is considered. A system of long-wave equations governing the deformations of the upper surface and the interface between the liquids is derived. The nonlinear simulations reveal the existence of numerous dynamical regimes, including two-dimensional stationary flows and standing waves, three-dimensional standing waves with different spatial periods, and three-dimensional travelling waves. The general diagram of the flow regimes is constructed.

The flow mechanisms of shape-changing moving bodies are investigated through the simple model of a foil that is rapidly retracted over a spanwise distance as it is towed at constant angle of attack. It is shown experimentally and through simulation that by altering the shape of the tip of the retracting foil, different shape-changing conditions may be reproduced, corresponding to: (i) a vanishing body, (ii) a deflating body and (iii) a melting body. A sharp-edge, ‘vanishing-like’ foil manifests strong energy release to the fluid; however, it is accompanied by an additional release of energy, resulting in the formation of a strong ring vortex at the sharp tip edges of the foil during the retracting motion. This additional energy release introduces complex and quickly evolving vortex structures. By contrast, a streamlined, ‘shrinking-like’ foil avoids generating the ring vortex, leaving a structurally simpler wake. The ‘shrinking’ foil also recovers a large part of the initial energy from the fluid, resulting in much weaker wake structures. Finally, a sharp edged but hollow, ‘melting-like’ foil provides an energetic wake while avoiding the generation of a vortex ring. As a result, a melting-like body forms a simple and highly energetic and stable wake, that entrains all of the original added mass fluid energy. The three conditions studied correspond to different modes of flow control employed by aquatic animals and birds, and encountered in disappearing bodies, such as rising bubbles undergoing phase change to fluid.

Under the action of solid body rotation, homogeneous neutrally stratified turbulence undergoes anisotropization and onset of the inverse energy cascade. These processes are investigated using a quasi-normal scale elimination (QNSE) theory in which successive coarsening of a flow domain yields scale-dependent eddy viscosity and diffusivity. The effect of rotation increases with increasing scale and manifests in anisotropization of the eddy viscosities, eddy diffusivities and kinetic energy spectra. Not only the vertical (in the direction of the vector of rotation $\unicode[STIX]{x1D734}$) and horizontal eddy viscosities and eddy diffusivities become different but, reflecting both directional and componental anisotropization, there emerge four different eddy viscosities. Three of them decrease relative to the eddy viscosity in non-rotating flows while one increases; the horizontal ‘isotropic’ viscosity decreases at the fastest rate. This behaviour is indicative of the increasing redirection of the energy flux to larger scales, the phenomenon that can be associated with the energy backscatter or inverse energy cascade. On scales comparable to the Woods’s scale which is the rotational analogue of the Ozmidov length scale in stably stratified flows, the horizontal viscosity rapidly decreases, and in order to keep it positive, a weak rotation limit is invoked. Within that limit, an analytical theory of the transition from the Kolmogorov to a rotation-dominated turbulence regime is developed. It is shown that the dispersion relation of linear inertial waves is unaffected by turbulence while all one-dimensional energy spectra undergo steepening from the Kolmogorov $-5/3$ to the $-3$ slope.

The impact of fluidic actuation on the wake and drag of a three-dimensional blunt body is investigated experimentally. Jets blowing tangentially to the main flow force the wake with variable frequency and amplitude. Depending on the forcing conditions, two flow regimes can be distinguished. First, in the case of broadband actuation with frequencies comprising the natural wake time scale, the convection of the jet structures enhances wake entrainment, shortens the length of the recirculating flow and increases drag. Secondly, at higher actuation frequencies, shear-layer deviation leads to fluidic boat tailing of the wake. It additionally lowers its turbulent kinetic energy thus reducing the entrainment of momentum towards the recirculating flow. The combination of both mechanisms produces a rise in the base pressure and reduces the drag of the model. Both actuation regimes are characterized by complementary velocity, pressure and drag measurements at several upstream conditions and control parameters. By adding curved surfaces to deviate the jets by the Coanda effect, periodic actuation is reinforced and drag reductions of approximately 20 % are achieved. The unsteady Coanda blowing not only intensifies the flow deviation and the base pressure recovery but also preserves the unsteady high-frequency forcing effect on the turbulent field. The present results encourage further development of fluidic control to improve the aerodynamics of road vehicles and provide a complementary insight into the relation between wake dynamics and drag.

Direct numerical simulations using up to $4096^{3}$ grid points on a deforming domain have been used to study the response of initially isotropic turbulence to a period of spatially uniform axisymmetric contraction (with one extensional and two equally compressive directions) and subsequent relaxation. A time-dependent strain rate is formulated to closely correspond to the downstream evolution in the wind tunnel experiments of Ayyalasomayajula & Warhaft (J. Fluid Mech., vol. 566, 2006, pp. 273–307), with a smoothly varying 4 : 1 contraction ratio. The application of strain leads to anisotropy in both the large scales and the small scales, in a manner where nonlinear effects not considered in rapid-distortion theory play an important role. Upon termination of strain, the small scales quickly return to isotropy while a residual level of anisotropy appears to persist at the large scales. The simulations are shown to reproduce many key findings from experiments, including distinctive changes in the form of the one-dimensional spectra in the extensional direction that arise at sufficiently high Reynolds number, during both the straining and relaxation periods. Scale-dependent measures of anisotropy are presented in terms of one-dimensional spectra and axisymmetric versions of the energy spectrum. To explain the observed changes in spectral shapes, various terms in the spectral evolution equation representing rapid pressure strain, slow pressure strain, production, nonlinear transfer and viscous dissipation are computed, showing that nonlinear effects take a dominant role when a wide range of scales exists. In particular, the ‘double-peak’ spectral form observed in experiments at high Reynolds number is found to be a consequence of the small scales relaxing towards isotropy much faster than the large scales. A comparison of results obtained from computational domains of varying sizes and grid resolutions show that the numerical findings are robust.

The shear rheology of an emulsion of viscous drops in the presence of finite inertia is investigated using direct numerical simulation. In the absence of inertia, emulsions display a non-Newtonian rheology with positive first and negative second normal stress differences. However, recently it was discovered that a small amount of drop-level inertia alters their signs – the first normal stress difference becomes negative and the second one becomes positive, each in a small range of capillary numbers (Li & Sarkar, J. Rheol., vol. 49, 2005, pp. 1377–1394). Sign reversal was shown numerically and analytically, but only in the limit of a dilute emulsion where drop–drop interactions were neglected. Here, we compute the rheology of a density- and viscosity-matched emulsion, accounting for the interactions in the volume fraction range of 5 %–27 % and Reynolds number range of 0.1–10. The computed rheological properties (effective shear viscosity and first and second normal stress differences) in the Stokes limit match well with previous theoretical (Choi–Schowalter in the dilute limit) and simulated results (for concentrated systems) using the boundary element method. The two distinct components of the rheology arising from the interfacial stresses at the drop surface and the perturbative Reynolds stresses are investigated as functions of the drop Reynolds number, capillary number and volume fraction. The sign change is caused by the increasing drop inclination in the presence of inertia, which in turn directly affects the interfacial stresses. Increase of the volume fraction or capillary number increases the critical Reynolds number for sign reversals due to enhanced alignment of the drops with the flow directions. The effect of increasing the volume fraction on the rheology is explained by relating it to interactions and specifically to the contact pair-distribution function computed from the simulation. The excess stresses are seen to show an approximately linear behaviour with the Reynolds number in the range of 0.1–5, while with the capillary number and volume fraction, the variation is weakly quadratic.

In this study we present a theoretical weakly nonlinear framework for the prediction of thermoacoustic oscillations close to Hopf bifurcations. We demonstrate the method for a thermoacoustic network that describes the dynamics of an electrically heated Rijke tube. We solve the weakly nonlinear equations order by order, discuss their contribution on the overall dynamics and show how solvability conditions at odd orders give rise to Stuart–Landau equations. These equations, combined together, describe the nonlinear dynamical evolution of the oscillations’ amplitude and their frequency. Because we retain the contribution of several acoustic modes in the thermoacoustic system, the use of adjoint methods is required to derive the Landau coefficients. The analysis is performed up to fifth order and compared with time domain simulations, showing good agreement. The theoretical framework presented here can be used to reduce the cost of investigating oscillations and subcritical phenomena close to Hopf bifurcations in numerical simulations and experiments and can be readily extended to consider, e.g. the weakly nonlinear interaction of two unstable thermoacoustic modes.

We report on an experimental study on the dynamics and fragmentation of water droplets levitated in a sound field exposed to a single laser-induced cavitation bubble. The nucleation of the cavitation bubble leads to a shock wave travelling inside the droplet and reflected from pressure release surfaces. Experiments and simulations study the location of the high negative pressures inside the droplet which result into secondary cavitation. Later, three distinct fragmentation scenarios are observed: rapid atomization, sheet formation and coarse fragmentation. Rapid atomization occurs when the expanding bubble, still at high pressure, ruptures the liquid film separating the bubble from the surrounding air and a shock wave is launched into the surrounding air. Sheet formation occurs due to the momentum transfer of the expanding bubble; for sufficiently small bubbles, the sheet retracts because of surface tension, while larger bubbles may cause the fragmentation of the sheet. Coarse fragmentation is observed after the first collapse of the bubble, where high-speed jets emanate from the surface of the droplet. They are the result of surface instability of the droplet combined with the impulsive pressure generated during collapse. A parameter plot for droplets in the size range between 0.17 and 1.5 mm and laser energies between 0.2 and 4.0 mJ allows the separation of these three regimes.