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The flow and the turbulence structure generated by a circular porous cylinder of diameter $D$ containing solid cylinders of diameter $d$ placed in an open channel of depth $h\approx 0.5D$ are investigated using eddy-resolving simulations which resolve the wakes past the individual solid cylinders in the array. The solid cylinders extend from the bed through the water surface. This geometrical set-up is directly relevant to understand the physics of flow past an emerged patch of aquatic vegetation developing in a river channel or over its floodplain. Simulations are conducted with different solid volume fractions (SVFs) of the porous cylinder ( $0.034<\text{SVF}<0.23$ ), relative diameters of the solid cylinders ( $d/D=0.03$ and 0.06) and with flat and equilibrium scour bathymetry corresponding to the start and respectively the end of the erosion and deposition process. Comparison with the limiting case of a solid cylinder ( $\text{SVF}=1$ ) is also discussed. The bed shear stress distributions and the turbulent flow fields are used to explain the sediment erosion mechanisms inside and around the porous cylinder. Simulations of the flat-bed cases reveal that for sufficiently large SVF values ( $\text{SVF}>0.2$ ), necklace vortices form around the upstream face of the cylinder, the downflow penetrates partially inside the porous cylinder and a region of strong flow acceleration forms on the sides of the porous cylinder. These flow features are used to explain the development of scour around high-SVF porous cylinders. The effects of the SVF and $d/D$ on generating ‘corridors’ of strong flow acceleration in between the solid cylinders and energetic eddies in the wake of these cylinders are discussed, as these flow features control the amplification of the bed shear stress inside the porous cylinder. Simulations results are also used to quantify the time-averaged drag forces on the cylinders in the array, to identify the regions where these forces are comparable to those induced on an isolated cylinder and the percentage of cylinders in the array subject to relatively large mean drag forces. A logarithmic decrease of the mean time-averaged streamwise drag coefficient of the solid cylinders, $\overline{C}_{d}$ , with increasing non-dimensional frontal area per unit volume of the porous cylinder, $aD$ , is observed. Behind the cylinder, the eddies shed in the separated shear layers (SSLs) of the porous cylinder, and, for sufficiently large SVFs, the von Kármán wake billows are the main coherent structures responsible for the amplification of the bed shear stress and sediment entrainment. This paper also analyses the vertical non-uniformity of the mean flow and turbulent kinetic energy, and discusses how the SVF and bathymetry affect the spatial extent of the wake region (e.g. length of the SSLs and steady wake, total wake length) and other relevant variables (e.g. strength of the bleeding flow, dominant wake frequencies, turbulence amplification in the near wake). For the relatively shallow flow conditions ( $D/h\approx 2.0$ ) considered, the simulation results show that the antisymmetric (von Kármán) shedding of wake billows behind the porous cylinder is greatly weakened once equilibrium scour conditions are approached. Comparison with data from laboratory experiments and from 3-D and 2-D simulations conducted for long porous cylinders (no bed) is also discussed.

To investigate which component of the anisotropic permeability tensor of porous media influences turbulence over porous walls, direct numerical simulation of anisotropic porous-walled channel flows is performed by the D3Q27 multiple-relaxation-time lattice Boltzmann method. The presently considered anisotropic permeable walls have square pore arrays aligned with the Cartesian axes. Vertical, streamwise and spanwise pore arrays are systematically introduced to the walls to impose anisotropic permeability. Simulations are carried out at a friction Reynolds number of 111 and 230, which is based on the averaged friction velocity of the porous bottom and the smooth top walls. It is found that streamwise and spanwise permeabilities enhance turbulence whilst vertical permeability itself does not. In particular, the enhancement of turbulence is remarkable over porous walls with streamwise permeability. Over streamwise permeable walls, development of high- and low-speed streaks is prevented whilst large-scale intermittent patched patterns of ejection motions are induced. It is revealed by two-point correlation analysis that streamwise permeability allows the development of streamwise large-scale perturbations induced by Kelvin–Helmholtz instability. Spectral analysis reveals that this perturbation contributes to the enhancement of the Reynolds shear stress, leading to significant skin friction of the porous interface. Through the comparison between the two different Reynolds-number cases, it is found that, as the Reynolds number increases, the streamwise perturbation becomes larger and more organized. Consequently, owing to the enhancement of the large-scale perturbation, a significant Reynolds-number dependence of the skin friction of the porous interface can be observed over the streamwise permeable wall. It is also implied that the wavelength of the perturbation can be reasonably scaled by the outer-layer length scale.

Wakes formed behind bluff bodies frequently reveal complex patterns of coherent vortical structures, with emergence of streamwise spatial periodicity particularly in the mid-wake region. In some cases, the vortex positions also maintain symmetry about the wake centreline. For the case in which two pairs of vortices are generated per shedding cycle, thereby constituting the so-called ‘2P’ mode wake, assumptions of spatial periodicity and symmetry allow for development of a mathematically tractable model using the point-vortex approximation. Our previous work (Basu & Stremler, Phys. Fluids, vol. 27 (10), 2015, 103603) considered staggered 2P wake configurations with two glide-reflective pairs of vortices shed in each period. Here we investigate the dynamics of a spatially periodic point-vortex street consisting of two pairs of vortices arranged with reflective symmetry about the streamwise centreline. Because of the symmetry, it is possible to model the spatially periodic point-vortex dynamics as an integrable Hamiltonian system. For a particular choice of initial condition, the topological structure of the Hamiltonian level curves is determined by location in a circulation–impulse parameter space. These Hamiltonian level curves delineate multiple regimes of motion, with all vortex motions within one regime being qualitatively identical. This approach thus enables identification and a full classification of all possible vortex motions in this constrained system. There also exist a limited number of equilibrium configurations with no relative vortex motion; some of these relative equilibria are neutrally stable to (appropriate) perturbations. Only one such neutrally stable equilibrium configuration continues to preserve the distinct four-vortex array, and numerical experiments indicate that these configurations are also neutrally stable to small perturbations that break the spatial symmetry. We apply this analysis to identify the parameter values necessary for co-existence of two closely spaced, neutrally stable Kármán vortex streets that preserve the assumed symmetry. Finally, comparison of the model dynamics to a wake pattern reported in the literature suggests that the classification of exotic wakes should be based on more details than just the number of vortices periodically shed by the body.

This paper revisits the problem of tidal conversion at a ridge in a uniformly stratified fluid of limited depth using measurements of complex-valued added mass. When the height of a sub-marine ridge is non-negligible with respect to the depth of the water, the tidal conversion can be enhanced in the supercritical regime or reduced in the subcritical regime with respect to the large depth situation. Tidal conversion can even be null for some specific cases. Here, we study experimentally the influence of finite depth on the added mass coefficients for three different ridge shapes. We first show that, at low forcing frequency, the tidal conversion is weakly enhanced by shallow depth for a semi-circular ridge. In addition, added mass coefficients measured for a vertical ridge show strong similarities with the ones obtained for the semi-circular ridge. Nevertheless, the enhancement of the tidal conversion at low forcing frequency for the vertical ridge has not been observed, in contrast with its supercritical shape. Finally, we provide the experimental evidence of a lack of tidal conversion due to the specific shape of a ridge for certain depth and frequency tuning.

A complete Hamiltonian formalism is suggested for inertial waves in rotating incompressible fluids. Resonance three-wave interaction processes – decay instability and confluence of two waves – are shown to play a key role in the weakly nonlinear dynamics and statistics of inertial waves in the rapid rotation case. Future applications of the Hamiltonian approach to inertial wave theory are investigated and discussed.

We present a single-layer feed-forward artificial neural network architecture trained through a supervised learning approach for the deconvolution of flow variables from their coarse-grained computations such as those encountered in large eddy simulations. We stress that the deconvolution procedure proposed in this investigation is blind, i.e. the deconvolved field is computed without any pre-existing information about the filtering procedure or kernel. This may be conceptually contrasted to the celebrated approximate deconvolution approaches where a filter shape is predefined for an iterative deconvolution process. We demonstrate that the proposed blind deconvolution network performs exceptionally well in the a priori testing of two-dimensional Kraichnan, three-dimensional Kolmogorov and compressible stratified turbulence test cases, and shows promise in forming the backbone of a physics-augmented data-driven closure for the Navier–Stokes equations.

Rotating Rayleigh–Bénard convection is typified by a variety of regimes with very distinct flow morphologies that originate from several instability mechanisms. Here we present results from direct numerical simulations of three representative set-ups: first, a fluid with Prandtl number $Pr=6.4$ , corresponding to water, in a cylinder with a diameter-to-height aspect ratio of $\unicode[STIX]{x1D6E4}=2$ ; second, a fluid with $Pr=0.8$ , corresponding to $\text{SF}_{6}$ or air, confined in a slender cylinder with $\unicode[STIX]{x1D6E4}=0.5$ ; and third, the main focus of this paper, a fluid with $Pr=0.025$ , corresponding to a liquid metal, in a cylinder with $\unicode[STIX]{x1D6E4}=1.87$ . The obtained flow fields are analysed using the sparsity-promoting variant of the dynamic mode decomposition (DMD). By means of this technique, we extract the coherent structures that govern the dynamics of the flow, as well as their associated frequencies. In addition, we follow the temporal evolution of single modes and present a criterion to identify their direction of travel, i.e. whether they are precessing prograde or retrograde. We show that for moderate $Pr$ a few dynamic modes suffice to accurately describe the flow. For large aspect ratios, these are wall-localised waves that travel retrograde along the periphery of the cylinder. Their DMD frequencies agree with the predictions of linear stability theory. With increasing Rayleigh number $Ra$ , the interior gradually fills with columnar vortices, and eventually a regular pattern of convective Taylor columns prevails. For small aspect ratios and close enough to onset, the dominant flow structures are body modes that can precess either prograde or retrograde. For $Pr=0.8$ , DMD additionally unveiled the existence of so far unobserved low-amplitude oscillatory modes. Furthermore, we elucidate the multi-modal character of oscillatory convection in low- $Pr$ fluids. Generally, more dynamic modes must be retained to accurately approximate the flow. Close to onset, the flow is purely oscillatory and the DMD reveals that these high-frequency modes are a superposition of oscillatory columns and cylinder-scale inertial waves. We find that there are coexisting prograde and retrograde modes, as well as quasi-axisymmetric torsional modes. For higher $Ra$ , the flow also becomes unstable to wall modes. These low-frequency modes can both coexist with the oscillatory modes, and also couple to them. However, the typical flow feature of rotating convection at moderate $Pr$ , the quasi-steady Taylor vortices, is entirely absent in low- $Pr$ flows.

Dense granular heap flows are common in nature, such as during avalanches and landslides, as well as in industrial flows. In granular heap flows, rapid flow is localized near the free surface with the thickness of the rapidly flowing layer dependent on the overall flow rate. In the region deep beneath the surface, exponentially decaying creeping flow dominates with characteristic decay length depending only on the geometry and not the overall flow rate. Existing continuum models for dense granular flow based upon local constitutive equations are not able to simultaneously predict both of these experimentally observed features – failing to even predict the existence of creeping flow beneath the surface. In this work, we apply a scale-dependent continuum approach – the non-local granular fluidity model – to steady, dense granular flows on a heap between two smooth, frictional side walls. We show that the model captures the salient features of both the flow-rate-dependent, rapidly flowing surface layer and the flow-rate-independent, slowly creeping bulk under steady flow conditions.

The consideration of viscoelasticity within fluid dynamical boundary element methods has traditionally required meshing over the whole flow domain. In turn, a major advantage of the boundary element method is lost, namely the need to consider only surface boundary integrals. Here, using a generalised reciprocal relation and viscoelastic force singularities, a boundary element method is developed for linear viscoelastic flows. We proceed to explore finite-deformation microswimming in a linear Maxwell fluid. We firstly deduce a finite-amplitude generalisation of a previously reported result that the flow field is unchanged between a Newtonian and linear Maxwell fluid for prescribed small-amplitude deformations. Hence Purcell’s theorem holds for a linear Maxwell fluid. We proceed to consider deformation swimming in a linear Maxwell fluid given an external forcing. Boundary scattering trajectories for an exemplar squirmer approaching a surface are observed to exhibit a weak dependence on the Deborah number, while the trajectories of a sperm and monotrichous bacterium near a surface are predicted to be essentially unaffected at moderate Deborah number. In turn, the latter supports the common simplification of using Newtonian Stokes flows for studying flagellate swimming in linear Maxwell media. In addition, the motion of a magnetic helix under the influence of an external magnetic field is considered, and highlights that linear viscoelasticity can significantly impact the propagation of the helix, in turn demonstrating that even linear rheology is important to consider for forced swimmers. Finally, the presented framework requires minimalistic adjustments to Newtonian boundary element codes, enabling rapid implementation, and is more generally applicable, for instance to studies of particle interactions in active linear rheology on the microscale.

We analyse the displacement of one fluid by a second immiscible fluid through a narrow channel of finite length which connects two reservoirs. We assume that the channel width slowly decreases in the direction of flow, and that the fluids have different viscosity and density. We examine the stability of the interface and find that there are Saffman–Taylor and Rayleigh–Taylor type modes, which may dominate in the narrow and wide regions of the channel, respectively. The gradient of the pressure jump across the interface associated with the surface tension acts to stabilise the interface, and for intermediate channel widths, this effect may dominate the destabilisation associated with both the Rayleigh–Taylor and Saffman–Taylor instabilities, provided the rate of change of the channel width with distance along the channel is sufficient. We also note that the effect of the converging channel leads to instability of long-wavelength modes owing to the quasi-static acceleration of the flow through the cell: we consider cases in which this effect only occurs at much lower wavenumbers than the most unstable Saffman–Taylor and Rayleigh–Taylor modes. We show that there is a maximum wavenumber for instability, which varies with position in the channel. By integrating the growth rate of each wavenumber in time as the interface moves across the channel, we predict the mode which grows to the greatest amplitude as the interface traverses the channel.

An experimental and numerical study was conducted to examine the effects of internal surface curvature and leading-edge angle on the shock waves and steady flow fields produced by axisymmetric ring wedges. Test models with leading-edge-radius-normalised internal radii of curvature of $R_{c}=\{1,1.5,2\}$ and leading-edge angles of $\unicode[STIX]{x1D6FC}=\{0^{\circ },4^{\circ },8^{\circ }\}$ were manufactured and tested. Experimental shadowgraph and schlieren results were obtained for Mach numbers ranging from 2.8 to 3.6 using a blowdown supersonic wind tunnel with accompanying numerical results for additional insight. The higher the internal surface curvature and leading-edge angle, the greater the flow fields were impacted. As a result, steeper compression waves were formed, thus curving the shock wave more noticeably. The internal surface curvature and leading-edge angle were both found to have an effect on the trailing-edge expansion fans. This altered the shape of downstream shock wave structures. The highest curvature models produced steady double reflection patterns due to the imposed internal surface curvature. The effects of conical and curved internal surfaces were explored for the presence of flow-normal curvature and the curving of the attached shock waves.

Consistent equations for turbulent open-channel flows on a smooth bottom are derived using a turbulence model of mixing length and an asymptotic expansion in two layers. A shallow-water scaling is used in an upper – or external – layer and a viscous scaling is used in a thin viscous – or internal – layer close to the bottom wall. A matching procedure is used to connect both expansions in an overlap domain. Depth-averaged equations are then obtained in the approximation of weakly sheared flows which is rigorously justified. We show that the Saint-Venant equations with a negligible deviation from a flat velocity profile and with a friction law are a consistent set of equations at a certain level of approximation. The obtained friction law is of the Kármán–Prandtl type and successfully compared to relevant experiments of the literature. At a higher precision level, a consistent three-equation model is obtained with the mathematical structure of the Euler equations of compressible fluids with relaxation source terms. This new set of equations includes shearing effects and adds corrective terms to the Saint-Venant model. At this level of approximation, energy and momentum resistances are clearly distinguished. Several applications of this new model that pertains to the hydraulics of open-channel flows are presented including the computation of backwater curves and the numerical resolution of the growing and breaking of roll waves.

Fully turbulent Taylor–Couette flow between independently rotating cylinders is investigated experimentally in a wide-gap configuration ( $\unicode[STIX]{x1D702}=0.5$ ) around the maximum transport of angular momentum. In that regime turbulent Taylor vortices are present inside the gap, leading to a pronounced axial dependence of the flow. To account for this dependence, we measure the radial and azimuthal velocity components in horizontal planes at different cylinder heights using particle image velocimetry. The ratio of angular velocities of the cylinder walls $\unicode[STIX]{x1D707}$ , where the torque maximum appears, is located in the low counter-rotating regime ( $\unicode[STIX]{x1D707}_{max}(\unicode[STIX]{x1D702}=0.5)=-0.2$ ). This point coincides with the smallest radial gradient of angular velocity in the bulk and the detachment of the neutral surface from the outer cylinder wall, where the azimuthal velocity component vanishes. The structure of the flow is further revealed by decomposing the flow field into its large-scale and turbulent contributions. Applying this decomposition to the kinetic energy, we can analyse the formation process of the turbulent Taylor vortices in more detail. Starting at pure inner cylinder rotation, the vortices are formed and strengthened until $\unicode[STIX]{x1D707}=-0.2$ quite continuously, while they break down rapidly for higher counter-rotation. The same picture is shown by the decomposed Nusselt number, and the range of rotation ratios, where turbulent Taylor vortices can exist, shrinks strongly in comparison to investigations at much lower shear Reynolds numbers. Moreover, we analyse the scaling of the Nusselt number and the wind Reynolds number with the shear Reynolds number, finding a communal transition at approximately $Re_{S}\approx 10^{5}$ from classical to ultimate turbulence with a transitional regime lasting at least up to $Re_{S}\geqslant 2\times 10^{5}$ . Including the axial dispersion of the flow into the calculation of the wind amplitude, we can also investigate the wind Reynolds number as a function of the rotation ratio $\unicode[STIX]{x1D707}$ , finding a maximum in the low counter-rotating regime slightly larger than $\unicode[STIX]{x1D707}_{max}$ . Based on our study it becomes clear that the investigation of counter-rotating Taylor–Couette flows strongly requires an axial exploration of the flow.

We have investigated thermoacoustically amplified quasi-planar nonlinear waves driven to the limit of shock-wave formation in a variable-area looped resonator geometrically optimized to maximize the growth rate of the quasi-travelling-wave second harmonic. Optimal conditions result in velocity leading pressure by approximately $40^{\circ }$ in the thermoacoustic core and not in pure travelling-wave phasing. High-order unstructured fully compressible Navier–Stokes simulations reveal three regimes: (i) modal growth, governed by linear thermoacoustics; (ii) hierarchical spectral broadening, resulting in a nonlinear inertial energy cascade, (iii) shock-wave-dominated limit cycle, where energy production is balanced by dissipation occurring at the captured shock-thickness scale. The acoustic energy budgets in regime (i) have been analytically derived, yielding an expression of the Rayleigh index in closed form and elucidating the effect of geometry and hot-to-cold temperature ratio on growth rates. A time-domain nonlinear dynamical model is formulated for regime (ii), highlighting the role of second-order interactions between pressure and heat-release fluctuations, causing asymmetry in the thermoacoustic energy production cycle and growth rate saturation. Moreover, energy cascade is inviscid due to steepening in regime (ii), with the $k$ th harmonic growing at $k/2$ -times the modal growth rate of the thermoacoustically sustained second harmonic. The frequency energy spectrum in regime (iii) is shown to scale with a $-5/2$ power law in the inertial range, rolling off at the captured shock-thickness scale in the dissipation range. We have thus shown the existence of equilibrium thermoacoustic energy cascade analogous to hydrodynamic turbulence.

The propagation of full-depth lock-exchange bottom gravity currents past a submerged array of circular cylinders is investigated using laboratory experiments and large eddy simulations. Firstly, to investigate the front velocity of gravity currents across the whole range of array density $\unicode[STIX]{x1D719}$ (i.e. the volume fraction of solids), the array is densified from a flat bed ( $\unicode[STIX]{x1D719}=0$ ) towards a solid slab ( $\unicode[STIX]{x1D719}=1$ ) under a particular submergence ratio $H/h$ , where $H$ is the flow depth and $h$ is the array height. The time-averaged front velocity in the slumping phase of the gravity current is found to first decrease and then increase with increasing $\unicode[STIX]{x1D719}$ . Next, a new geometrical framework consisting of a streamwise array density $\unicode[STIX]{x1D707}_{x}=d/s_{x}$ and a spanwise array density $\unicode[STIX]{x1D707}_{y}=d/s_{y}$ is proposed to account for organized but non-equidistant arrays ( $\unicode[STIX]{x1D707}_{x}\neq \unicode[STIX]{x1D707}_{y}$ ), where $s_{x}$ and $s_{y}$ are the streamwise and spanwise cylinder spacings, respectively, and $d$ is the cylinder diameter. It is argued that this two-dimensional parameter space can provide a more quantitative and unambiguous description of the current–array interaction compared with the array density given by $\unicode[STIX]{x1D719}=(\unicode[STIX]{x03C0}/4)\unicode[STIX]{x1D707}_{x}\unicode[STIX]{x1D707}_{y}$ . Both in-line and staggered arrays are investigated. Four dynamically different flow regimes are identified: (i) through-flow propagating in the array interior subject to individual cylinder wakes ( $\unicode[STIX]{x1D707}_{x}$ : small for in-line array and arbitrary for staggered array; $\unicode[STIX]{x1D707}_{y}$ : small); (ii) over-flow propagating on the top of the array subject to vertical convective instability ( $\unicode[STIX]{x1D707}_{x}$ : large; $\unicode[STIX]{x1D707}_{y}$ : large); (iii) plunging-flow climbing sparse close-to-impermeable rows of cylinders with minor streamwise intrusion ( $\unicode[STIX]{x1D707}_{x}$ : small; $\unicode[STIX]{x1D707}_{y}$ : large); and (iv) skimming-flow channelized by an in-line array into several subcurrents with strong wake sheltering ( $\unicode[STIX]{x1D707}_{x}$ : large; $\unicode[STIX]{x1D707}_{y}$ : small). The most remarkable difference between in-line and staggered arrays is the non-existence of skimming-flow in the latter due to the flow interruption by the offset rows. Our analysis reveals that as $\unicode[STIX]{x1D719}$ increases, the change of flow regime from through-flow towards over- or skimming-flow is responsible for increasing the gravity current front velocity.

Fluid flows in nature and applications are frequently subject to periodic velocity modulations. Surprisingly, even for the generic case of flow through a straight pipe, there is little consensus regarding the influence of pulsation on the transition threshold to turbulence: while most studies predict a monotonically increasing threshold with pulsation frequency (i.e. Womersley number, $\unicode[STIX]{x1D6FC}$ ), others observe a decreasing threshold for identical parameters and only observe an increasing threshold at low $\unicode[STIX]{x1D6FC}$ . In the present study we apply recent advances in the understanding of transition in steady shear flows to pulsating pipe flow. For moderate pulsation amplitudes we find that the first instability encountered is subcritical (i.e. requiring finite amplitude disturbances) and gives rise to localized patches of turbulence (‘puffs’) analogous to steady pipe flow. By monitoring the impact of pulsation on the lifetime of turbulence we map the onset of turbulence in parameter space. Transition in pulsatile flow can be separated into three regimes. At small Womersley numbers the dynamics is dominated by the decay turbulence suffers during the slower part of the cycle and hence transition is delayed significantly. As shown in this regime thresholds closely agree with estimates based on a quasi-steady flow assumption only taking puff decay rates into account. The transition point predicted in the zero $\unicode[STIX]{x1D6FC}$ limit equals to the critical point for steady pipe flow offset by the oscillation Reynolds number (i.e. the dimensionless oscillation amplitude). In the high frequency limit on the other hand, puff lifetimes are identical to those in steady pipe flow and hence the transition threshold appears to be unaffected by flow pulsation. In the intermediate frequency regime the transition threshold sharply drops (with increasing $\unicode[STIX]{x1D6FC}$ ) from the decay dominated (quasi-steady) threshold to the steady pipe flow level.

A new macroscopic model describing the rheology and microstructure of dilute emulsions with droplet morphology is developed based on an internal contravariant conformation tensor variable which is physically identified with the deformed ellipsoidal geometry of the dispersed phase. The model is consistent with existing first-order capillary number, $O(Ca)$ , theory describing the microstructure as well as $O(Ca^{2})$ theory describing the emulsion-contributed extra stress. These asymptotic solutions are also used to determine all of the model parameters, making it the only macroscopic emulsion model that is consistent with all available asymptotic theories in the limit of small $Ca$ . The governing equations are obtained from the Poisson and dissipation brackets, as developed for an incompressible fluid system endowed with an internal contravariant second-order tensor, subject to the imposition of the constraint of a unit determinant. First proposed by Maffettone & Minale (J. Non-Newtonian Fluid Mech., vol. 78, 1998, pp. 227–241), this constraint physically corresponds to conservation of the volume of the dispersed phase in the emulsion. The Hamiltonian of the emulsion is expressed through the surface energy of the dispersed phase, in addition to the kinetic energy, following previous work by Grmela et al. (J. Non-Newtonian Fluid Mech., vol. 212, 2014, pp. 1–12), but employing a more accurate evaluation of the surface area in terms of the internal contravariant conformation tensor. Structural predictions of the ellipsoid droplet morphology obtained with the new model are compared with classic experiments by Torza et al. (J. Colloid Interface Sci., vol. 38, 1972, pp. 395–411), showing good agreement.

When a gas bubble grows by diffusion in a gas–liquid solution, it affects the distribution of gas in its surroundings. If the density of the solution is sensitive to the local amount of dissolved gas, there is the potential for the onset of natural convection, which will affect the bubble growth rate. The experimental study of the successive quasi-static growth of many bubbles from the same nucleation site described in this paper illustrates some consequences of this effect. The enhanced growth due to convection causes a local depletion of dissolved gas in the neighbourhood of each bubble beyond that due to pure diffusion. The quantitative data of sequential bubble growth provided in the paper show that the radius-versus-time curves of subsequent bubbles differ from each other due to this phenomenon. A simplified model accounting for the local depletion is able to collapse the experimental curves and to predict the progressively increasing bubble detachment times.

Differential heating applied at a single horizontal boundary forces ‘horizontal convection’, even when there is no net heat flux through the boundary. However, almost all studies of horizontal convection have been limited to a special class of problem in which temperature or heat flux differences were applied in only one direction and over the horizontal length of a box (the Rossby problem; Rossby, Deep-Sea Res., vol. 12, 1965, pp. 9–16). These conditions strongly constrain the flow. Here we report laboratory experiments and direct numerical simulations (DNS) extending the results of Griffiths & Gayen (Phys. Rev. Lett., vol. 115, 2015, 204301) for horizontal convection forced by boundary conditions imposed in a two-dimensional periodic array at a horizontal boundary. The experiments use saline and freshwater fluxes at a permeable base with the imposed boundary salinity having a horizontal length scale one quarter of the width of the box. The flow reaches a state in which the net boundary buoyancy flux vanishes and the bulk of the fluid shows an inertial range of turbulence length scales. A regime transition is seen for increasing water depth, from an array of individual coherent plumes on the forcing scale to convection dominated by emergent larger scales of overturning. The DNS explore the analogous thermally forced case with sinusoidal boundary temperature of wavenumber $n=4$ , and are used to examine the Rayleigh number ( $Ra$ ) dependence for shallow- and deep-water cases. For shallow water the flow transitions with increasing $Ra$ from laminar to turbulent boundary layer regimes that are familiar from the Rossby problem and which have normalised heat transport scaling as $Nu\sim Ra^{1/5}$ and $Nu\sim (Ra\,Pr)^{1/5}$ , with $Nu$ the Nusselt number and $Pr$ the Prandtl number, in this case maintaining a stable array of coherent turbulent plumes. For deep-water and large $Ra$ the laminar scaling transitions to $Nu\sim (Ra\,Pr)^{1/4}$ , with the scales of turbulence extending to the dimensions of the box. The $1/4$ power law regime is explained in terms of the momentum of symmetric, inviscid large scales of motion in the interior coupled to diffusive loss of heat through stabilised parts of the boundary layer. The turbulence production is predominantly by shear instability rather than convection, with viscous dissipation distributed throughout the bulk of the fluid. These conditions are not seen in the highly asymmetric flow in the Rossby problem even at Rayleigh numbers up to six orders of magnitude greater than the transition found here. The new inertial interior regime has the rate of supply of available potential energy, and its removal by mixing of density, increasing as $Ra^{5/4}$ , which is faster than $Ra^{6/5}$ in the Rossby problem. Irreversible mixing is confined close to the forcing boundary and is very much larger than the viscous dissipation, which is proportional to $Ra$ .

The interaction of stationary streaks undergoing non-modal growth with modally unstable instability waves in a supersonic flat-plate boundary-layer flow is studied using numerical computations. For incompressible flows, previous studies have shown that boundary-layer modulation due to streaks below a threshold amplitude level can stabilize the Tollmien–Schlichting instability waves, resulting in a delay in the onset of laminar–turbulent transition. In the supersonic regime, the most-amplified linear waves become three-dimensional, corresponding to oblique, first-mode waves. This change in the character of dominant instabilities leads to an important change in the transition process, which is now dominated by oblique breakdown via nonlinear interactions between pairs of first-mode waves that propagate at equal but opposite angles with respect to the free stream. Because the oblique breakdown process is characterized by a strong amplification of stationary streamwise streaks, artificial excitation of such streaks may be expected to promote transition in a supersonic boundary layer. Indeed, suppression of those streaks has been shown to delay the onset of transition in prior literature. This paper investigates the nonlinear evolution of initially linear optimal disturbances that evolve into finite-amplitude streaks in a two-dimensional, Mach 3 adiabatic flat-plate boundary-layer flow, followed by the modal instability characteristics of the perturbed, streaky boundary-layer flow. Both parts of the investigation are performed with the plane-marching parabolized stability equations. Consistent with previous findings, the present study shows that optimally growing stationary streaks can destabilize the first-mode waves, but only when the spanwise wavelength of the instability waves is equal to or smaller than twice the streak spacing. Transition in a benign disturbance environment typically involves first-mode waves with significantly longer spanwise wavelengths, and hence, these waves are stabilized by the optimal growth streaks. Thus, as long as the amplification factors for the destabilized, short wavelength instability waves remain below the threshold level for transition, a significant net stabilization is achieved, yielding a potential transition delay that may be comparable to the length of the laminar region in the uncontrolled case.