The internal flow within an evaporating sessile droplet has intriguing fluid mechanics important to various microfluidics applications. In the present study, a phenomenon is observed through numerical methods wherein the buoyancy-driven flow structure inside a droplet on a non-wetting substrate transitions from an axisymmetric toroidal vortex flow to a non-axisymmetric single vortex flow with increase in the substrate temperature. As the axisymmetric nature of the droplet flow field and evaporation characteristics are broken, the internal velocity accelerates significantly. The transition, which is attributed to a flow instability inside the droplet, is more prone to occur as the droplet volume or the contact angle increases. The onset of the flow transition is analysed as the amplification of a small perturbation, thereby establishing a correlation between the flow instability and the Rayleigh number (Ra). Specifically, when Ra exceeds some critical value, the onset of the flow transition is observed, which explains the effects of substrate temperature and droplet volume on the internal flow. Next, the influence of the droplet contact angle on the critical Ra was investigated, and the underlying reasons were analysed. Finally, we discuss the heat transfer efficiency within the droplet and analyse why the internal flow tends to transition to a non-axisymmetric flow pattern from an energy minimization perspective.

An experimental study has been conducted on the near wake of a 6 : 1 spheroid, in both uniform and stratified backgrounds. The pitch angle, $\theta$, was varied from $0^\circ \text { to }20^\circ$. When $\theta = 0^\circ$, stratification decreases the characteristic wake element spacing so a characteristic Strouhal number ($St$) increases from 0.32 to 0.4. However, a similar measure scaled on wake momentum thickness shows the wake spacing to converge on those measured for other bluff and streamlined bodies. There is an apparent effect of Reynolds number, which changes the location of separation lines and hence the initial wake thickness. When $\theta > 0^\circ$, the wake is a combination of the usual drag wake together with a collection of streamwise vortices that have separated from the body, and this wake geometry can evolve in ways that are measurably different from the zero incidence case. These differences may be limited to the near wake, as the later evolution appears to converge with previous bluff- and streamlined bodies, with normalised wake height, $L_V = 0.5$ and centreline velocity, $\bar {u}_0 = 0.3$ at $Nt = 10$, as the early wake enters the non-equilibrium regime with similar values to previously studied stratified wakes. In the presence of density stratification, the inclined wake itself generates large-scale internal wave undulations with time scale $2{\rm \pi} /N$, even when the background stratification is not strong and a body-based Froude number is $O(10)$. The geometry and strengths of the primary streamwise vortices are not symmetric, mirroring previous results from experiments and computations in the literature.

Turbulent flow around curved tandem cylinders has been studied for the first time, by means of direct numerical simulation. The convex configuration was used, with a nominal gap ratio of $L/D = 3$ and a Reynolds number of 3900. Along the span, the flow regimes vary from alternating overshoot/reattachment to co-shedding. Three distinct Strouhal numbers coexist in the flow that are tied directly to different tandem cylinder flow regimes. This result differs substantially from convex curved tandem cylinders at a transitional Reynolds number, where only a single dominant frequency is found. All regimes exhibit some degree of instability, so that the flow can be considered multistable. A mode switch from alternating overshoot/reattachment to symmetric reattachment is found. Complex interactions are observed between the primary instability, the shear layer instability and the flow mode alterations. As opposed to previous investigations with single and tandem straight cylinders in the subcritical flow regime, our results indicate that there may be direct feedback from the primary instability to the shear layer instability. The downdraft region in the gap exhibits slow meandering, and may travel upstream and amplify the shear layer instability, causing early transition in the gap shear layer. This downdraft is governed by the slow modulations of the vortex formation region in the lower gap, meaning that the vortex dynamics of this region may indirectly influence the shear layer instability higher up in the gap.

Two-dimensional (2-D) quadrant analysis is generally used for investigating flow and sediment dynamics around a rigid structure in open channel flows. Given that particle distribution around rigid obstacles is not spatially uniform and changes in time, while vortices evolve to become three-dimensional (3-D) structures, 2-D quadrant analysis might be unsuitable to completely determine the sediment transport. Hence, 3-D quadrant and 3-D octant analyses should be considered, using the 3-D instantaneous velocity data and relative 3-D bursting process to define sediment transport surrounding the submerged square and circular cylinders. The turbulent kinetic energy (TKE), transition probabilities, occurrence probabilities, stress fraction and angles of inclination of 3-D bursting events are considered to quantify the coherent structures surrounding the cylinders and their interaction with bed particles. Experiments were conducted at the Hydraulics and Water Resources Engineering Laboratory, School of Infrastructure, Indian Institute of Technology, Bhubaneswar, Odisha, India, and velocity data were recorded at different cross-sections around the submerged cylinders using an acoustic Doppler velocimeter. Results show that the TKE is greater for internal ejection, external ejection, and internal sweep, external sweep in the upstream of the circular and square cylindrical structures. On the other part, the TKE is significant for internal ejection, external ejection, and internal sweep, external sweep in the downstream of the aligned square cylindrical structures, which justifies the highest scour depth that occurred upstream of the circular and square cylindrical structures and downstream of the aligned square cylindrical structure. The transition probability of the bursting events was determined using the Markov process from the measured velocity data to investigate the consecutive occurrence of bursting events. Further, the importance of sweeps and ejections on sediment erosion surrounding the cylinders within the scour hole at various stages of its development was investigated via 3-D quadrant analysis of the bursting occurrences. The outcomes show that external sweep and internal ejection events are active mechanisms for bed particle transport surrounding the cylinder. The maximum transition probability values are found around aligned cylindrical structures in comparison with the circular and square cylindrical structures in the transverse direction. This depicts the formation of a trailing vortex on both sides of the aligned square cylindrical object. The results reveal that the effect of inclination angles with respect to the water flow is greater for internal ejection and external sweep from upstream to downstream within the scour hole surrounding the cylindrical structures at various phases of development as horseshoe vortices and downflow develop upstream of the cylindrical structures while trailing vortices and wake vortices form at the top and downstream of the cylindrical structures. Internal and external ejection have a higher stress fraction than an internal and external sweep for square cylinders with alignment angles of 0°, 20° and circular cylinders over underdeveloped and developed scoured beds, respectively. With the higher percentage of fractional contributions for internal sweeps, the external sweep is predicted close to the cylindrical objects in comparison with the internal ejection and external ejection events because of the formation and warping of the horseshoe vortex close to the cylindrical objects, suggesting a significant probability of 3-D bursting occurrences with sediment movement near the cylindrical structures.

The Marangoni flow induced by an insoluble surfactant on a fluid–fluid interface is a fundamental problem investigated extensively due to its implications in colloid science, biology, the environment and industrial applications. Here, we study the limit of a deep liquid subphase with negligible inertia (low Reynolds number, $Re\ll {1}$), where the two-dimensional problem has been shown to be described by the complex Burgers equation. We analyse the problem through a self-similar formulation, providing further insights into its structure and revealing its universal features. Six different similarity solutions are found. One of the solutions includes surfactant diffusion, whereas the other five, which are identified through a phase-plane formalism, hold only in the limit of negligible diffusion (high surface Péclet number $Pe_s\gg {1}$). Surfactant ‘pulses’, with a locally higher concentration that spreads outward, lead to two similarity solutions of the first kind with a similarity exponent $\beta =1/2$. On the other hand, distributions that are locally depleted and flow inwards lead to similarity of the second kind, with two different exponents that we obtain exactly using stability arguments. We distinguish between ‘dimple’ solutions, where the surfactant has a quadratic minimum and $\beta =2$, from ‘hole’ solutions, where the concentration profile is flatter than quadratic and $\beta =3/2$. Each of these two cases exhibits two similarity solutions, one valid prior to a critical time $t_*$ when the derivative of the concentration is singular, and another one valid after $t_*$. We obtain all six solutions in closed form, and discuss predictions that can be extracted from these results.

It is well known that buoyancy suppresses, and can even laminarise, turbulence in upward heated pipe flow. Heat transfer seriously deteriorates in this case. A new direct numerical simulation model is established to simulate flow-dependent heat transfer in an upward heated pipe. The model shows good agreement with experimental results. Three flow states are simulated for different values of the buoyancy parameter $C$: shear turbulence, laminarisation and convective turbulence. The latter two regimes correspond to the heat transfer deterioration regime and the heat transfer recovery regime, respectively (Jackson & Li 2002; Bae et al., Phys. Fluids, vol. 17, issue 10, 2005; Zhang et al., Appl. Energy, vol. 269, 2020, 114962). We confirm that convective turbulence is driven by a linear instability (Su & Chung, J. Fluid Mech., vol. 422, 2000, pp. 141–166) and that the deteriorated heat transfer within convective turbulence is related to a lack of rolls near the wall, which leads to weak mixing between the flow near the wall and the centre of the pipe. Having surveyed the fundamental properties of the system, we perform a nonlinear non-modal stability analysis, which seeks the minimal perturbation that triggers a transition from the laminar state. Given the differences between shear and convective turbulence, we aim to determine how the nonlinear optimal (NLOP) changes as the buoyancy parameter $C$ increases. We find that at first, the NLOP becomes thinner and closer to the wall. Most importantly, the critical initial energy $E_0$ required to trigger turbulence keeps increasing, implying that attempts to trigger it artificially may not be an efficient means to improve heat transfer at larger $C$. At $C=6$, a new type of NLOP is discovered, capable of triggering convective turbulence from lower energy, but over a longer time. It is active only in the centre of the pipe. We next compare the transition processes, from linear instability and by the nonlinear non-modal excitation. At $C=4$, linear instability leads to a state that approaches a travelling wave solution or periodic solutions, while the minimal seed triggers shear turbulence before decaying to convective turbulence. Deeper into the parameter space for convective turbulence, at $C=6$, the new nonlinear optimal triggers convective turbulence directly. Detailed analysis of the periodic solution at $C=4$ reveals three stages: growth of the unstable eigenfunction, the formation of streaks, and the decay of the streaks. The stages of the cycle correspond to changes in the linear instability of the turbulent mean velocity profile. Unlike the self-sustaining process for classical shear flows, where the streak is disrupted via instability, here, decay of the streak is more closely linked to suppression of the linear instability of the mean flow, and hence suppression of the rolls. Flow visualisations at $C$ up to $10$ also show similar processes, suggesting that the convective turbulence in the heat transfer recovery regime is sustained by these three typical processes.

Time-resolved two-dimensional two-component particle image velocimetry measurements with high spatial resolution are carried out in a water tank agitated by four blades rotating at constant speed. Different blade geometries and rotation speeds are tested for the purpose of modifying turbulent flow conditions. In all cases where no baffles are used to break the rotation, the Zeman length is an order of magnitude smaller than the Taylor length. Compared with the cases with baffles which break the rotation, in the unbaffled cases turbulence production and/or mean advection are significant and the turbulence nonlinearity is dramatically reduced for the horizontal (i.e. normal to the axis of rotation) two-point turbulence fluctuating velocities. This nonlinearity reduction is manifest not only in the interscale turbulent energy transfer but also in the interspace turbulent energy transfer, which nearly vanishes. However, the nonlinearity is not reduced for the vertical two-point turbulence fluctuating velocities: the corresponding interscale turbulent transfer rate is in fact intensified, and its dependence on the two-point separation distance, as well as that of the corresponding interspace turbulent transfer rate, which does not vanish, is significantly modified. Even though non-homogeneities are very different for different blades and rotation speeds in the unbaffled cases, the horizontal fluctuating velocity's second-order structure function collapses with scalings which resemble predictions for homogeneous turbulence subject to strong rotation. The vertical fluctuating velocity's second-order structure function does not collapse for different blade geometries by neither these nor the Kolmogorov predictions.

The present work focuses on a specific bouncing behaviour as a spherical particle settles through a density interface in the absence of a neutral buoyant position. This behaviour was initially discovered by Abaid et al. (Phys. Fluids, vol. 16, issue 5, 2004, pp. 1567–1580) in salinity-induced stratification. Both experimental and numerical investigations are conducted to understand this phenomenon. In our experiments, we employ particle image velocimetry (PIV) to measure the velocity distribution around the particle and to capture the transient wake structure. Our findings reveal that the bouncing process begins after the wake detaches from the particle. The PIV results indicate that an upward jet forms at the central axis behind the particle following wake detachment. By performing a force decomposition procedure, we quantify the contributions from the buoyancy of the wake ($F_{sb}$) and the flow structure ($F_{sj}$) to the enhanced drag. It is observed that $F_{sb}$ contributes primarily to the enhanced drag at the early stage, whereas $F_{sj}$ plays a critical role in reversing the particle's motion. Furthermore, our results indicate that the jet is a necessary condition for the occurrence of the bouncing motion. We also explore the minimum velocities (where negative values denote the occurrence of bouncing) of the particle, while varying the lower Reynolds number $Re_l$, the Froude number $Fr$, and the upper Reynolds number $Re_u$, within the ranges $1 \leqslant Re_l\leqslant 125$, $115 \leqslant Re_u\leqslant 356$ and $2 \leqslant Fr\leqslant 7$. Our findings suggest that the bouncing behaviour is influenced primarily by $Re_l$. Specifically, we observe that the bouncing motion occurs below a critical lower Reynolds number $Re^\ast _{l}=30$ in our experiments. In the numerical simulations, the highest value for this critical number is $Re^\ast _{l}=46.2$, which is limited to the parametric ranges studied in this work.

We report a numerical investigation of a previously noticed but less explored flow state transition in two-dimensional turbulent Rayleigh–Bénard convection. The simulations are performed in a square domain over a Rayleigh number range of $10^7 \leq Ra \leq 2 \times 10^{11}$ and a Prandtl number range of $0.25 \leq Pr \leq 20$. The transition is characterized by the emergence of multiple satellite eddies with increasing $Ra$, which orbit around and interact with the main vortex roll in the system. Consequently, the main roll is squeezed to a smaller size compared with the domain and wanders around in the bulk region irregularly and extensively. This is in sharp contrast to the flow state before the transition, which is featured by a domain-sized circulatory roll with its vortex centre ‘condensed’ near the domain's centre. Detailed velocity field analysis reveals that there exists an abrupt increase in the energy fluctuations of the Fourier modes during the transition. Based on this phase-transition-like signal, the critical condition for the transition is found to follow a scaling relation as $Ra_t \sim Pr^{1.41}$ where $Ra_t$ is the critical Rayleigh number for the transition. This scaling relation is quantitatively explained by a phenomenological model grounded on the bistability behaviour (i.e. spontaneous and stochastic switching between the two flow states) observed at the edge of the transition. The model can also account for the effects of aspect ratio on the transition reported in the literature (van der Poel et al., Phys. Fluids, vol. 24, 2012).

The particle trajectories in irrotational, incompressible and inviscid deep-water surface gravity waves are open, leading to a net drift in the direction of wave propagation commonly referred to as the Stokes drift, which is responsible for catalysing surface wave-induced mixing in the ocean and transporting marine debris. A balance between phase-averaged momentum density, kinetic energy density and vorticity for irrotational, monochromatic and spatially periodic two-dimensional water waves is derived by working directly within the Lagrangian reference frame, which tracks particle trajectories as a function of their labels and time. This balance should be expected as all three of these quantities are conserved following particles in this system. Vorticity in particular is always conserved along particles in two-dimensional inviscid flow, and as such even in its absence it is the value of the vorticity that fundamentally sets the drift, which in the Lagrangian frame is identified as the phase-averaged momentum density of the system. A relationship between the drift and the geometric mean water level of particles is found at the surface, which highlights connections between the geometry and dynamics. Finally, an example of an initially quiescent fluid driven by a wavelike pressure disturbance is considered, showing how the net momentum and energy from the surface pressure disturbance transfer to the wave field, and recognizing the source of the mean Lagrangian drift as the net momentum required to generate an irrotational surface wave by any conservative force.

A phenomenological model is proposed to estimate the initial thickness of the liquid microlayer forming beneath a vapour bubble growing on a solid surface upon nucleate boiling. The model employs an analogy between the microlayer formation and the classic plate withdrawal problem. It calculates the microlayer thickness by considering it as a Landau–Levich film, where the thickness is a function of the meniscus speed and radius of curvature. Given the nearly hemispherical shape of the bubble during the early growth stage when the microlayer is first deposited, we assume that the meniscus speed can be approximated by the bubble expansion rate, and estimate the meniscus curvature using the Rayleigh equations. Unlike previous theories that assume that the bubble radius growth is proportional to the square root of time, the proposed model does not rely on any specific law of growth for vapour bubbles. The model is validated for predicting the microlayer thickness in water and ethanol, showing good agreement with experimental measurements and empirical correlations. Subsequent analyses of the microlayer interface profile address inconsistent reports – some described a wedge-like shape, whereas others reported a slight outward curvature with decreasing thickness in the outer region. This discrepancy is attributed to a reduction in the expansion rate of the microlayer's outer edge, particularly when the bubble reaches its maximum width. Our model provides insights into microlayer dynamics, essential to boiling heat transfer, as the evaporative heat flux through the microlayer is very sensitive to its initial thickness.

When studying two-dimensional fluid–body interactions in the low-Froude limit, traditional asymptotic theory predicts a waveless free surface at every order. The waves are, in fact, exponentially small and it has been well-established that such waves ‘switch on’ seemingly instantaneously across so-called Stokes lines, partitioning the fluid domain into wave-free regions and regions with waves. In three dimensions, the Stokes-line concept extends to higher-dimensional Stokes surfaces. This article is concerned with the archetypal problem of uniform flow over a point source, reminiscent of, but separate to, the famous Kelvin wave problem. In prior research, the intersection of the Stokes surface with the free surface was found, in implicit form, for this case of a point source. However, on account of the algebraic manipulations required, it is not clear how this approach can be extended to more challenging settings. Here we develop a numerical-based procedure that allows the Stokes surface to be computed. The intersections of the Stokes surfaces with both the free surface and the deeper fluid are discussed for the case of the point source. Crucially, this procedure provides an important tool for generalising exponential asymptotics to the case of nonlinear (non-point-source) wave-generating bodies.

We use large-eddy simulations to study the penetration of a buoyant plume carrying a passive tracer into a stably stratified layer with constant buoyancy frequency. Using a buoyancy-tracer volume distribution, we develop a method for objectively partitioning plume fluid in buoyancy-tracer space into three regions, each of which corresponds to a coherent region in physical space. Specifically, we identify a source region where undiluted plume fluid enters the stratified layer, a transport region where much of the transition from undiluted to mixed fluid occurs in the plume cap and an accumulation region corresponding to a radially spreading intrusion. This method enables quantification of different measures of turbulence and mixing within each of the three regions, including potential energy and turbulent kinetic energy dissipation rates, an activity parameter and the instantaneous mixing efficiency. We find that the most intense buoyancy gradients lie in a thin layer at the cap of the penetrating plume. This provides the primary stage of mixing between plume and environment and exhibits a mixing efficiency around 50 %. Newly generated mixtures of environmental and plume fluid join the intrusion and experience relatively weak turbulence and buoyancy gradients. As the intrusion spreads radially, environmental fluid surrounding the intrusion is mixed into the intrusion with moderate mixing efficiency. This dominates the volume of environmental fluid entrained into the region containing plume fluid. However, the ‘strongest’ entrainment, as measured by the specific entrainment rate, is largest in the plume cap, where the most buoyant environmental fluid is entrained.