The rise of thermals in the atmosphere has attracted a lot of attention since the early work of Morton et al. (Proc. R. Soc. Lond. A, vol. 234, 1956, pp. 1–23), who proposed that entrainment into a thermal was proportional to the surface area of the thermal and to the mean vertical velocity of the thermal. This paper presents new analytical solutions for the heights of rise of buoyant thermals in both stably and unstably stratified environments, for both negatively and positively buoyant sources, and where the sources have different size and strength (momentum) characteristics. The limiting cases of these analytical solutions are consistent with previous work. These analytical solutions do not appear elsewhere, and provide a compact set of equations that are easy to apply to a wide range of circumstances. The solutions are dependent upon the entrainment hypothesis, which is of course only an approximation, but the simplicity of the analytical solutions allows easy calculation and additional insights. These include the fact that while heights of rise are strongly dependent on both source strength and size for flows in stable environments, the dilution at the top of rise is independent of the source momentum. Further, in a stable environment, there is a conserved quantity that has dimensions proportional to vertical momentum. For negatively buoyant flows in an unstably stratified environment, thermals having low initial momentum will reach a maximum height, while thermals with high initial momentum will entrain sufficient buoyant environmental fluid that they will eventually become positively buoyant and continue to rise indefinitely.

Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers (${Ra}_{cr}$) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The ${Ra}_{cr}$ value depends on the non-dimensional frequency $\omega$ of the boundary heat-flux modulation. Floquet theory is used to find ${Ra}_{cr}$ for linear stability, and the energy method is used to find ${Ra}_{cr}$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $\omega$, with only the latter at large $\omega$. For a given frequency, the linear stability ${Ra}_{cr}$ is generally higher than the nonlinear stability ${Ra}_{cr}$, as expected. For large $\omega$, ${Ra}_{cr} \omega ^{-2}$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $\omega$. The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.

We introduce a new approach for modelling the swelling, drying and elastic behaviour of hydrogels, which leverages the tractability of classical linear-elastic theory whilst incorporating nonlinearities arising from large swelling strains. Relative to a reference state of a fully swollen gel, in which the polymer scaffold may only comprise less than $1\,\%$ of the total volume, a constitutive model for the Cauchy stress tensor is presented, which linearises around small deviatoric strains corresponding to shearing deformations of the material whilst allowing for a nonlinear relation between stress and isotropic strains. Such isotropic strains are considered only to be a consequence of losses and gains of water, while the hydrogel is taken to be instantaneously incompressible. The dynamics governing swelling and drying are described by coupling the interstitial flow of the water through the porous gel with the elastic response of the gel. This approach allows for a complete description of gel behaviour using only three macroscopic polymer-fraction-dependent parameters: an osmotic modulus, a shear modulus and a permeability. It is shown how these three material parameters can, in principle, be determined experimentally using a simple rheometry experiment in which a gel is compressed between two plates surrounded by water and the total force on the top plate is measured. To illustrate our approach, we solve for the swelling of a gel under horizontal confinement and for a partially dried hydrogel bead placed in water.

Measurements of fluctuating wall pressure in a high-Reynolds-number flow over a body of revolution are described. With a strong axial pressure gradient and moderate lateral curvature, this non-equilibrium flow is relevant to marine applications as well as short-haul urban transportation. The wall-pressure spectrum and its scaling are discussed, along with its relation to the space–time structure. As the flow decelerates downstream, the root-mean-square level of the pressure drops together with the wall shear stress ($\tau _w$) and is consistently approximately 7$\tau _w$. While the associated dimensional spectra see a broadband reduction of over 15 dB per Hz, they appear to attain a single functional form, collapsing to within 2 dB when normalized with the wall-wake scaling where $\tau _w$ is the pressure scale and $U_e/\delta$ is the frequency scale. Here, $\delta$ is the boundary layer thickness and $U_e$ is the local free-stream velocity. The general success of the wall-wake scaling, including in the viscous $f^{-5}$ region, suggests that the large-scale motions in the outer layer play a predominant role in the near-wall turbulence and wall pressure. On investigating further, we find that the instantaneous wall-pressure fluctuations are characterized by a quasi-periodic feature that appears to convect downstream at speeds consistent with the outer peak in the turbulence stresses. The conditional structure of this feature, estimated through peak detection in the time series, resembles that of a roller, supporting the embedded shear layer hypothesis (Schatzman & Thomas, J. Fluid Mech., vol. 815, 2017, pp. 592–642; Balantrapu et al., J. Fluid Mech., vol. 929, 2021, A9). Therefore, the outer-region shear-layer-type motions may be important when devising strategies for flow control, drag and noise reduction for decelerating boundary layers.

Effects of surface tension reduction on wind-wave growth are investigated using direct numerical simulations of air–water two-phase turbulent flows. The incompressible Navier–Stokes equations for air and water sides are solved using an arbitrary Lagrangian–Eulerian method with boundary-fitted moving grids. The wave growth of finite-amplitude and non-breaking gravity–capillary waves, whose wavelength is less than 0.07 m, is simulated for two cases of different surface tensions under a low-wind-speed condition of several metres per second. The results show that significant wave height for the smaller surface tension case increases faster than that for the larger surface tension case. Energy fluxes for gravity and capillary wave scales reveal that when the surface tension is reduced, the energy transfer from the significant gravity waves to capillary waves decreases, and the significant waves accumulate more energy supplied by wind. This results in faster wave growth for the smaller surface tension case. The effect on the scalar transfer across the air–water interface is also investigated. The results show that the scalar transfer coefficient on the water side decreases due to the surface tension reduction. The decrease is caused by suppression of turbulence in the water side. In order to support the conjecture, the surface tension effect is compared with laboratory experiments in a small wind-wave tank.

It has been reported that, in submarine landslides down a slope, only a small part of the landslide momentum was transferred to the tsunami while most was lost in the further propagation over the flat bed. The high momentum of landslides off a slope meant a long run-out and a great impact on underwater infrastructure. Nevertheless, little attention has been paid to the variation of momentum of deformable submarine landslides off a slope, i.e. the loss of momentum when the slides flow away from the slope over a flat bottom. In this paper, the translational momentum of deformable submarine granular landslides running down a non-erodible inclined bed is investigated with a two-phase smoothed particle hydrodynamics model. After flowing down the slope, the transport rate and flux of the landslide translational momentum along the propagation over the flat bottom are examined. The effects of physical variables of the slide, particularly the grain size, the initial compaction and the front intrusion angle on the variation of the translational momentum, are explored. Accordingly, scaling relations of the spatial-temporal maximum transport rate and flux of the landslide translational momentum, as well as those of the slide final run-out and the generated leading wave height, are proposed. These scaling relations, although based on numerical data of small-scale granular landslides, demonstrate a preliminary attempt to develop practical expressions in the framework of momentum for estimating the run-out of real submarine landslides and the impact on underwater infrastructure.

We investigate the role of the laminar/turbulent interface in the interscale energy transfer in a boundary layer undergoing bypass transition with the aid of the Kármán–Howarth–Monin–Hill (KHMH) equation. A local binary indicator function is used to detect the interface and employed subsequently to define two-point intermittencies. These are used to decompose the standard-averaged interscale and interspace energy fluxes into conditionally averaged components. We find that the inverse cascade in the streamwise direction reported in an earlier work arises due to events across the downstream or upstream interfaces of a turbulent spot. However, the three-dimensional energy flux maps reveal significant differences between these two regions: in the downstream interface, inverse cascade is stronger and dominant over a larger range of streamwise and spanwise separations. We explain this finding by considering a propagating spot of simplified shape as it crosses a fixed streamwise location. We derive also the conditionally averaged KHMH equation, thus generalising similar equations for single-point statistics to two-point statistics. We compare the three-dimensional maps of the conditionally averaged production and total energy flux within turbulent spots against the maps of standard-averaged quantities within the fully turbulent region. The results indicate remarkable dynamical similarities between turbulent spots and the fully turbulent region for two-point statistics. This has been known only for single-point quantities, and we demonstrate here that the similarity extends to two-point quantities as well.

Numerical simulations in two space dimensions are used to examine the dynamics, transport and equilibrium behaviours of a neutrally buoyant circular object immersed in an active suspension within a larger circular container. The continuum model of Gao et al. (Phys. Rev. Fluids, vol. 2, issue 9, 2017, 093302) represents the suspension of non-interacting, immotile, extensor-type microscopic agents that have a direction and strength, and align in response to strain rate. Such a suspension is well known to be unstable above an activity strength threshold, which depends upon the length scale of the confinement. Introducing the object leads to additional phenomenology. It can confine fluid between it and the container wall, which suppresses local suspension activity. However, its motion also correlates strain rates near its surface, which induce a correspondingly correlated active-stress response. Depending on the suspension activity strength, these mechanisms lead to either an attraction toward or a repulsion away from the container wall. In addition, a persistent propagating behaviour is found for modest activity strength, which provides a mechanism for long-range transport. When activity is so weak that the mobility of the object is essential to support suspension instability and sustain flow, all flow terminates when its mobility is diminished as it nears the container wall. If activity strength is scheduled in time, then these mechanisms could be used to perform relatively complex tasks with simple active agents.

The flow of three non-Newtonian fluids, comprising polymer and surfactant additives, in a periodically constricted tube (PCT) are experimentally compared. The radius of the tube walls is sinusoidal with respect to the streamwise direction. The three fluids are aqueous solutions of flexible polymers, rigid biopolymers and surfactants, which are typically used for drag-reduction in turbulent flows. Steady shear viscosity measurements demonstrate that rigid and flexible polymer solutions are shear-thinning, while surfactant solutions have a Newtonian and water-like shear viscosity. Capillary driven extensional rheology demonstrates that only flexible polymer solutions produce elastocapillary thinning. Particle shadow velocimetry is used to measure the velocity of each flow within the PCT at five Reynolds numbers spanning roughly 0.5 to 300. Relative to the Newtonian flows, rigid polymer solutions exhibit a blunt velocity profile. Flexible polymer solutions demonstrate a distinct chevron-shaped velocity contour and zones of opposing vorticity when the Deborah number exceeds 0.1. Using the vorticity transport equation, it is revealed that the opposing vorticity zones are coupled with a non-Newtonian torque. The PCT reveals that the surfactant solutions have similar non-Newtonian features as flexible polymer solutions – those being a chevron velocity pattern, opposing vorticity and a finite non-Newtonian torque. This observation is of practical importance since conventional shear and extensional rheometric measurements are not capable of demonstrating non-Newtonian features of the surfactant solutions. The investigation demonstrates that the PCT serves as a viable geometry for showing the non-Newtonian traits of dilute surfactant solutions.

This study analyses the stability characteristics of the shear layer vortices (SLV) in a reacting jet in crossflow, analysing effects of flame position, momentum flux ratio ($J$) and density ratio ($S$). It utilizes 40 kHz particle image velocimetry to characterize the dominant SLV frequencies, streamwise evolution and convective/global stability characteristics for three different canonical configurations, one non-reacting and two reacting (‘R1’ and ‘R2’). In the non-reacting case, both convective and global instability is observed, depending upon $S$ and $J$. Qualitatively similar $S$ dependencies occur for the R1 reacting case where the radial flame position lies outside the jet shear layer, albeit with slower SLV growth rates. When the flame lies inside the jet shear layer, the R2 reacting case, a qualitatively different behaviour is observed, as vorticity concentration in the shear layers is suppressed almost completely. Finally, we show that frequency and stability characteristics of the non-reacting and R1 cases can be scaled in a unified manner using a counter-current shear layer model. This model relates these SLV behaviours to a vorticity layer thickness, a velocity scale and an effective density ratio (noting that there are three distinct densities associated with the jet, the crossflow and the burned gases). These parameters were extracted from the data and used to collapse the frequency scaling, and to explain the transition to self-excited oscillatory behaviour.

The vortex shedding topology of a heavy pendulum oscillating in a dense fluid is investigated using time-resolved three-dimensional particle tracking velocimetry (tr-3-D-PTV). A series of experiments with eight different solid to fluid mass ratios $m^*$ in the range $[1.14, 14.95]$ and corresponding Reynolds numbers of up to $Re \sim O(10^4)$ was conducted. The period of oscillation depends heavily on $m^*$. The relation between amplitude decay and oscillation frequency is non-monotonic, having a damping optimum at $m^* \approx 2.50$. Moreover, a novel digital object tracking (DOT) method using vorticity-magnitude iso-surfaces is implemented to analyse vortical structures. A similar vortex shedding topology is observed for various mass ratios $m^*$. Our observations show that first, a vortex ring in the pendulum's wake is formed. Soon after, the initial ring breaks down to two clearly distinguishable structures of similar size. One of the two vortices remains on the circular path of the pendulum, while the other detaches, propagates downwards, and eventually dissipates. The time when the first vortex is shed, and its initial propagation velocity, depend on $m^*$ and the momentum imparted by the spherical bob. The findings further show good agreement between the experimentally determined vortex shedding frequency and the theoretical vortex shedding time scale based on the Strouhal number.

Interaction of supercritical granular flow with obstacles in a confined channel generates shock waves characterised by a nearly parabolic front of agitated grains in the outer region and a heap of static grains in the inner region. The inner static heap results from granular collapse due to high volume fraction and enhanced collision rate near the obstacle. The present work reports interesting flow structures when granular shock waves are formed on an array of three identical triangular obstacles placed in a rectangular channel at different spacings. It is observed that spacing has a profound influence, resulting in three types of flow structures. Through dimensional analysis, it is found that the normalised shock stand-off distance primarily depends on the Froude number, $Fr$, and the normalised spacing between the wedges. The experimental data show a strong dependence on these parameters. The normalised shock stand-off distance decreases linearly for small $Fr$ and asymptotically approaches a small value at high $Fr$. The presence of a new stagnant dome-like structure results in a non-intuitive behaviour of shock stand-off with the wedge spacing. These features are discussed in detail using high-resolution shadowgraphy and the velocity field from particle image velocimetry.

This work presents the first experimental characterization of the flow field in the vicinity of periodically spaced discrete roughness elements (DRE) in a swept wing boundary layer. The time-averaged velocity fields are acquired in a volumetric domain by high-resolution dual-pulse tomographic particle tracking velocimetry. Investigation of the stationary flow topology indicates that the near-element flow region is dominated by high- and low-speed streaks. The boundary layer spectral content is inferred by spatial fast Fourier transform (FFT) analysis of the spanwise velocity signal, characterizing the chordwise behaviour of individual disturbance modes. The two signature features of transient growth, namely algebraic growth and exponential decay, are identified in the chordwise evolution of the disturbance energy associated with higher harmonics of the primary stationary mode. A transient decay process is instead identified in the near-wake region just aft of each DRE, similar to the wake relaxation effect previously observed in two-dimensional boundary layer flows. The transient decay regime is found to condition the onset and initial amplitude of modal crossflow instabilities. Within the critical DRE amplitude range (i.e. affecting boundary layer transition without causing flow tripping) the transient disturbances are strongly receptive to the spanwise spacing and diameter of the elements, which drive the modal energy distribution within the spatial spectra. In the super-critical amplitude forcing (i.e. causing flow tripping) the near-element stationary flow topology is dominated by the development of a high-speed and strongly fluctuating region closely aligned with the DRE wake. Therefore, elevated shears and unsteady disturbances affect the near-element flow development. Combined with the harmonic modes transient growth these instabilities initiate a laminar streak structure breakdown and a bypass transition process.

This paper presents and compares two different approaches to solving the problem of wave propagation across a large finite periodic array of surface-piercing vertical barriers. Both approaches are formulated in terms of a pair of integral equations, one exact and based on a spacing $\delta > 0$ between adjacent barriers and the other approximate and based on a continuum model formally developed by using homogenisation methods for small $\delta$. It is shown that the approximate method is simpler to evaluate than the exact method which requires eigenvalues and eigenmodes related to propagation in an equivalent infinite periodic array of barriers. In both methods, the numerical effort required to solve problems is independent of the size of the array. The comparison between the two methods allows us to draw important conclusions about the validity of homogenisation models of plate array metamaterial devices. The practical interest in this problem stems from the result that for an array of barriers there exists a critical value of radian frequency, $\omega _c$, dependent on $\delta$, below which waves propagate through the array and above which it results in wave decay. When $\delta \to 0$, the critical frequency is given by $\omega _c = \sqrt {g/d}$, where $d$ is the plate submergence and $g$ is the acceleration due to gravity, which relates to the resonance in narrow channels and is an example of local resonance, studied extensively in metamaterials. The results have implications for proposed schemes to harness energy from ocean waves and other problems related to rainbow trapping and rainbow reflection.

In this paper, we put forward a hypothesis for turbulent kinetic energy, Reynolds stresses and scalar variance in wall-bounded turbulent flows, whereby these quantities, when normalized with the kinematic viscosity, mean turbulent energy dissipation rate and scalar dissipation rate, are independent of the Reynolds and Péclet numbers when they are sufficiently large. In particular, there exist two scaling ranges: (i) an inertial-convective range at sufficiently large distance from the wall over which a $2/3$ power-law scaling emerges for all quantities mentioned above; (ii) a viscous-convective range between the viscous-diffusive and inertial-convective ranges at large Prandtl number over which the normalized scalar variance is constant. The relatively large amount of available wall turbulence data either provides reasonably good support for this hypothesis or at least exhibits a trend that is consistent with the predictions of this hypothesis. The relationship between the proposed scaling and the traditional wall scaling is discussed. Possible ultimate statistical states of wall turbulence are also proposed.

To explore the direction of inter-scale transfer of scalar variance between subgrid scale (SGS) and resolved scalar fields, direct numerical simulation data obtained earlier from two complex-chemistry lean hydrogen–air flames are analysed by applying Helmholtz–Hodge decomposition (HHD) to the simulated velocity fields. Computed results show backscatter of scalar (combustion progress variable $c$) variance, i.e. its transfer from SGS to resolved scales, even in a highly turbulent flame characterized by a unity-order Damköhler number and a ratio of Kolmogorov length scale to thermal laminar flame thickness as low as 0.05. Analysis of scalar fluxes associated with the solenoidal and potential velocity fields yielded by HHD shows that the documented backscatter stems primarily from the potential velocity perturbations generated due to dilatation in instantaneous local flames, with the backscatter being substantially promoted by a close alignment of the spatial gradient of mean scalar progress variable and the potential-velocity contribution to the local SGS scalar flux. The alignment is associated with the fact that combustion-induced thermal expansion increases local velocity in the direction of $\boldsymbol {\nabla } c$. These results call for development of SGS models capable of predicting backscatter of scalar variance in turbulent flames in large eddy simulations.

The instability mechanism behind a geometrically simple cylindrical roughness element continues to be a challenging topic in fluid mechanics. Considerable progress has been made in understanding the phenomena in recent years, but more research is needed to predict the temporal nature and spatial structure of the dominant instability in a given flow configuration. This is of particular interest, as these instabilities dictate the transition to turbulence and thus are significant for large-scale effects such as skin friction drag. A smoke-flow visualization study with a large variation of parameters, featuring a cylindrical roughness element connected to a linear traverse, has been performed. Results show good agreement with previous investigations and provide further insights into the stability properties, revealing several unexpected effects. For a low roughness aspect ratio $\eta$, no global instability is detected even at the highest roughness Reynolds number $Re_{kk}$, whereas a high aspect ratio indicates a delay in the onset of instability. From the acquired visualizations, we constructed the, so far, richest instability diagram of the wake behind an isolated roughness element in the $Re_{kk}\unicode{x2013}\eta$ space, sampled in the same measurement campaign. Furthermore, information regarding the dominant frequency in the wake can be extracted from the visualization images. Our results suggest a new scaling of the frequency as the velocity is increased. Finally, it is shown that the dominant frequency in a certain flow regime can be well predicted using a Strouhal number based on the cylinder diameter and the roughness velocity.

Large-eddy simulation is used to simulate the flow around a 6 : 1 prolate spheroid at $10^\circ$ and $20^\circ$ angles of attack, and Reynolds number $4.2 \times 10^6$. Flows with and without trip are compared to understand the relative effect of the trip on the state of the boundary layer and separation. For the tripped case, the geometry of the trip is resolved to better predict its effect on the downstream flow. The simulations employ overset grids that allow adequate resolution of the trips without significant increase in the overall computational cost. Results suggest that while the trip accelerates transition to turbulence at $10^\circ$, it does not induce a fully developed turbulent boundary layer as intended at $20^\circ$. Rather, the influence of the trip is localized, and the near-wall flow converges towards a solution similar to that of the non-tripped case upstream of separation. This is due to two distinct phenomena: directly downstream of the trip, favourable pressure gradient and streamline curvature effects suppress the disturbance on the windward side. Further along the spheroid, the boundary layer receives a small fraction of the initial perturbation due to spanwise and wall-normal streamline curvatures inducing a secondary flow that advects the low-momentum trip wake to the leeward side. The locations of transition and separation are insensitive to the presence of the trip. The simulation results are used to construct a regime map that identifies different regions characterized by distinct boundary layer properties and flow features. The present results underscore the difficulty associated with tripping smooth bodies at angle of attack, and the importance of accounting for transition in simulations of such flows, even on tripped geometries.

Vortex rings are critical for thrust production underwater. In the ocean, self-propelled mesozooplankton generate vortices while swimming within a weakly stratified fluid. While large-scale biogenic transport has been observed during vertical migration in the wild and lab experiments, little focus has been given to the evolution of induced vortex rings as a function of their propagation direction relative to the density gradient. In this study, the evolution of an isolated vortex ring crossing the interface of a stable two-layer system is examined as a function of its translation direction with respect to gravity. The vortex ring size and position are visualized using planar laser-induced fluorescence (PLIF) and the induced vorticity field derived from particle image velocimetry (PIV) is examined. It is found that the production of baroclinic vorticity significantly affects the propagation of vortex rings crossing the density interface. As a result, any expected symmetry between vortex rings travelling from dense to light fluids and from light to dense fluids breaks down. In turn, the maximum penetration depth of the vortex ring occurs in the case in which the vortex propagates against the density gradient due to the misalignment of the pressure and density gradients. Our results have far-reaching implications for the characterization of local ecosystems in marine environments.