We investigate theoretically the steady incompressible viscoelastic flow in a rigid axisymmetric cylindrical pipe with varying cross-section. We use the Oldroyd-B viscoelastic constitutive equation to model the fluid viscoelasticity. First, we derive exact general formulae: for the total average pressure-drop as a function of the wall shear rate and the viscoelastic axial normal extra-stress; for the viscoelastic extra-stress tensor and the Trouton ratio as functions of the fluid velocity on the axis of symmetry; and for the viscoelastic extra-stress tensor along the wall in terms of the shear rate at the wall. Then we exploit the classic lubrication approximation, valid for small values of the square of the aspect ratio of the pipe, to simplify the original governing equations. The final equations are solved analytically using a regular perturbation scheme in terms of the Deborah number, De, up to eighth order in De. For a hyperbolically shaped pipe, we reveal that the reduced pressure-drop and the Trouton ratio can be recast in terms of a modified Deborah number, Dem, and the polymer viscosity ratio, η, only. Furthermore, we enhance the convergence and accuracy of the eighth-order solutions by deriving transformed analytical formulae using Padé diagonal approximants. The results show the decrease of the pressure drop and the enhancement of the Trouton ratio with increasing Dem and/or increasing η. Comparison of the transformed solutions with numerical simulations of the lubrication equations using pseudospectral methods shows excellent agreement between the results, even for high values of Dem and all values of η, revealing the robustness, validity and efficiency of the theoretical methods and techniques developed in this work. Last, it is shown that the exact solution for the Trouton ratio gives a well-defined and finite solution for any value of Dem and reveals the reason for the failure of the corresponding high-order perturbation series for Dem > 1/2.

Wave-assisted propulsion (WAP) systems directly convert wave energy into thrust using elastically mounted hydrofoils. The wave conditions as well as the design of the hydrofoil drive the fluid–structure interaction of the hydrofoil and, consequently, its performance. We employ simulations using a sharp-interface immersed boundary method to examine the effect of three key parameters on the flow physics, the fluid–structure interaction as well as thrust performance of these systems – the stiffness of the torsional spring, the location of the pitch axis and the Strouhal number. We demonstrate the utility of ‘maps’ of energy exchange between the flow and the hydrofoil system, as a way to understand and predict these characteristics. The force-partitioning method (FPM) is used to decompose the pressure forces into interpretable components and to quantify the mechanisms associated with thrust generation. Based on the results from FPM, a phenomenological model for the thrust generated by the WAP foil is presented. The parameters associated with this model are estimated based on data from over 450 distinct simulations. The predictions of the model are compared with the simulations and the use of this model for guiding WAP design is discussed.

Microorganism motility often takes place within complex, viscoelastic fluid environments, e.g. sperm in cervicovaginal mucus and bacteria in biofilms. In such complex fluids, strains and stresses generated by the microorganism are stored and relax across a spectrum of length and time scales and the complex fluid can be driven out of its linear response regime. Phenomena not possible in viscous media thereby arise from feedback between the swimmer and the complex fluid, making swimming efficiency co-dependent on the propulsion mechanism and fluid properties. Here, we parameterize a flagellar motor and filament properties together with elastic relaxation and nonlinear shear-thinning properties of the fluid in a computational immersed boundary model. We then explore swimming efficiency, defined as a particular flow rate divided by the torque required to spin the motor, over this parameter space. Our findings indicate that motor efficiency (measured by the volumetric flow rate) can be boosted or degraded by relatively moderate or strong shear thinning of the viscoelastic environment.

Turbulent entrainment at the turbulent/non-turbulent interface (TNTI) plays an important role in understanding the turbulent diffusion. While entrainment in fully developed canonical turbulent flows has been extensively studied, the evolution of entrainment in spatially developing flows remains poorly understood. In this work, characteristics of entrainment and the effect of vortices on entrainment of the shear layer separated from a wall-mounted fence are studied by the experiment in a water channel. The shedding vortex experiences a series of stages, including generation, growth, deformation and breakdown into smaller vortices. With the development of the flow, entrainment varies correspondingly. The prograde vortex near the TNTI is found to suppress entrainment but have little effect on the detrainment process, while the retrograde vortex promotes entrainment and suppresses detrainment as well. Consequently, the local entrainment velocity is decreased by the prograde vortex and increased more significantly by the retrograde vortex. Along the streamwise direction, the time-mean entrainment velocity is smallest where the prograde vortex is strongest in the vortex deformation stage. However, the largest time-mean entrainment velocity is located where the enstrophy gradient near the TNTI is greatest after reattachment, rather than where the retrograde vortex is strongest shortly after the breakdown of the shedding vortex, because the scarcity of retrograde vortices in the vicinity of the TNTI makes their long-time cumulative contribution not as significant as their local enhancement. The present study reveals how entrainment evolves in the separated and reattaching flow, and improves our understanding of the effect of vortices on entrainment.

Supersonic internal flows often exhibit multiple reflected shocks within a limited distance. These shocks can interact with each other in a complex manner due to the characteristics of the shock wave–turbulent boundary layer interaction (STBLI), including flow distortion and the relaxing boundary layer. This study aims to characterise this type of interaction and to clarify its fluid physics. A separated STBLI zone was established either upstream or downstream, and another weaker STBLI was established in the opposing position to serve as a perturbation. Time-resolved measurements were employed to characterise the mean separation and unsteadiness as the two regions approached each other, as well as their relationship. The experimental results indicated that the STBLI could affect the separation and reattachment of the other STBLI through either the decelerated or relaxing boundary layer. Despite a small deflection angle, the incident shock can amplify the low-frequency oscillations in the downstream STBLI region. Additionally, the interaction in the downstream region can be influenced by both low- and high-frequency oscillations associated with the upstream STBLI through a relaxing boundary layer. Despite the limited correlation observed between the low-frequency fluctuations in the downstream region and the boundary layer flow not far upstream, there still exists some degree of correlation between the low-frequency shock motions even when they are widely separated. Both the ‘upstream mechanism’ and ‘downstream mechanism’ have been observed, and the significance of low-frequency dynamics in the separated flow, relative to that of the upstream flow, is closely associated with interaction intensity.

In convergent geometry, the effect of convergence and compression on the Rayleigh–Taylor instability (RTI) and Richtmyer–Meshkov instability (RMI) modifies the growth rate and behaviour of the instabilities. In order to better understand how compression/expansion caused by axial strain rates (i.e. strain rates normal to the interface) change the instability dynamics, axial strain rates are applied to RMI in planar geometry, isolated from the effects of convergence. Potential flow theory for the linear regime shows the growth rate of the instability is modified to include the background velocity difference of the instability's width. Resolved two-dimensional simulations of single-mode RMI showed the potential flow model is accurate whilst the amplitude is small compared with the wavelength. The application of strain rate to an RMI-induced mixing layer was investigated using three-dimensional implicit large eddy simulations (ILES) of the quarter-scale $\theta$-group case by Thornber et al. (Phys. Fluids, vol. 29, 2017, 105107). Whilst the background strain rate contributed to the mixing layer's growth, it was to a smaller extent than expected. The shear production of axial turbulent kinetic energy from the strain rate modified the rate of bulk entrainment, affecting the mixing layer's growth and mixedness, such that the strained simulations no longer attained the same self-similar state. The capability of the buoyancy-drag model by Youngs & Thornber (Physica D, vol. 410, 2020, 132517) to predict the mixing layer width was investigated, using a model calibrated to the unstrained case. New terms were introduced into the buoyancy-drag model, which correspond to the shear production of turbulent kinetic energy.

This study explores heat and turbulent modulation in three-dimensional multiphase Rayleigh–Bénard convection using direct numerical simulations. Two immiscible fluids with identical reference density undergo systematic variations in dispersed-phase volume fractions, $0.0 \leq \varPhi \leq 0.5$, and ratios of dynamic viscosity, $\lambda _{\mu }$, and thermal diffusivity, $\lambda _{\alpha }$, within the range $[0.1\unicode{x2013}10]$. The Rayleigh, Prandtl, Weber and Froude numbers are held constant at $10^8$, $4$, $6000$ and $1$, respectively. Initially, when both fluids share the same properties, a 10 % Nusselt number increase is observed at the highest volume fractions. In this case, despite a reduction in turbulent kinetic energy, droplets enhance energy transfer to smaller scales, smaller than those of single-phase flow, promoting local mixing. By varying viscosity ratios, while maintaining a constant Rayleigh number based on the average mixture properties, the global heat transfer rises by approximately 25 % at $\varPhi =0.2$ and $\lambda _{\mu }=10$. This is attributed to increased small-scale mixing and turbulence in the less viscous carrier phase. In addition, a dispersed phase with higher thermal diffusivity results in a 50 % reduction in the Nusselt number compared with the single-phase counterpart, owing to faster heat conduction and reduced droplet presence near walls. The study also addresses droplet-size distributions, confirming two distinct ranges dominated by coalescence and breakup with different scaling laws.

In transonic flow conditions, buffeting associated with finite-amplitude lift fluctuations can limit the operational envelope of an aircraft. For both airfoils and wings, these oscillations have been linked to global flow instabilities that arise from a Hopf bifurcation. We employ a combination of numerical simulations and global stability analysis to investigate the near-critical behaviour of the oscillatory buffet-onset instability on airfoils. The flow is governed by the unsteady Reynolds-averaged Navier–Stokes equations, with a basic state provided by a steady-state solution. In the weakly nonlinear formulation, the disturbance amplitude is described by the Landau equation. The linear growth rate can be determined from either the simulations or the stability analysis, and the Landau constant is derived from simulations resulting in finite-amplitude equilibrium states. The results show that the Landau constant is nearly independent of Mach number and angle of attack for a given airfoil. Using the Landau constant derived from a small number of simulations, the stability analysis can be employed to efficiently capture the essential finite-amplitude behaviour needed to estimate the buffet-onset boundary. The stability analysis is shown to capture the envelope of lift oscillations during a continuous pitch of an airfoil, from pre-buffet through post-buffet lift levels.

In confined systems, the entrapment of a gas volume with an equivalent spherical diameter greater than the dimension of the channel can form extended bubbles that obstruct fluid circuits and compromise performance. Notably, in sealed vertical tubes, buoyant long bubbles cannot rise if the inner tube radius is below a critical value near the capillary length. This critical threshold for steady ascent is determined by geometric constraints related to the matching of the upper cap shape with the lubricating film surrounding the elongated part of the bubble. Developing strategies to overcome this threshold and release stuck bubbles is essential for applications involving narrow liquid channels. Effective strategies involve modifying the matching conditions with an external force field to facilitate bubble ascent. However, it is unclear how changes in acceleration conditions affect the motion onset of buoyancy-driven long bubbles. This study investigates the mobility of elongated bubbles in sealed tubes with an inner radius near the critical value inhibiting bubble motion in a vertical setting. Two strategies are explored to tune bubble motion, leveraging variations in axial and transversal accelerations: tube rotation around its axis and tube inclination relative to gravity. By revising the geometrical constraints of the simple vertical setting, the study predicts new thresholds based on rotational speed and tilt angle, respectively, providing forecasts for the bubble rising velocity under modified apparent gravity. Experimental measurements of motion threshold and rising velocity compare well with theoretical developments, thus suggesting practical approaches to control and tune bubble motion in confined environments.

Aqueous foams coarsen with time due to gas diffusion through the liquid between the bubbles. The mean bubble size grows, and small bubbles vanish. However, coarsening is little understood for foams with an intermediate liquid content, particularly in the presence of surfactant-induced attractive forces between the bubbles, measured by the interface contact angle where thin films meet the bulk liquid. Rigorous bubble growth laws have yet to be developed, and the evolution of bulk foam properties is unclear. We present a quasistatic numerical model for coarsening in two-dimensional wet foams, focusing on growth laws and related bubble properties. The deformation of bubble interfaces is modelled using a finite-element approach, and the gas flow through both films and Plateau borders is approximated. We give results for disordered two-dimensional wet foams with $256$ to $1024$ bubbles, at liquid fractions from $2\,\%$ to $25\,\%$, beyond the zero-contact-angle unjamming transition, and with contact angles up to $10^\circ$. Simple analytical models for the bubble pressures, film lengths and coarsening growth rates are developed to aid interpretation. If the contact angle is non-zero, we find that a prediction of the coarsening rate approaches a non-zero value as the liquid fraction is increased. We also find that an individual bubble's effective number of neighbours determines whether it grows or shrinks to a good approximation.

We describe the rising trajectory of bubbles in isotropic turbulence and quantify the slowdown of the mean rise velocity of bubbles with sizes within the inertial subrange. We perform direct numerical simulations of bubbles, for a wide range of turbulence intensity, bubble inertia and deformability, with systematic comparison with the corresponding quiescent case, with Reynolds number at the Taylor microscale from 38 to 77. Turbulent fluctuations randomise the rising trajectory and cause a reduction of the mean rise velocity $\tilde {w}_b$ compared with the rise velocity in quiescent flow $w_b$. The decrease in mean rise velocity of bubbles $\tilde {w}_b/w_b$ is shown to be primarily a function of the ratio of the turbulence intensity and the buoyancy forces, described by the Froude number $Fr=u'/\sqrt {gd}$, where $u'$ is the root-mean-square velocity fluctuations, $g$ is gravity and $d$ is the bubble diameter. The bubble inertia, characterised by the ratio of inertial to viscous forces (Galileo number), and the bubble deformability, characterised by the ratio of buoyancy forces to surface tension (Bond number), modulate the rise trajectory and velocity in quiescent fluid. The slowdown of these bubbles in the inertial subrange is not due to preferential sampling, as is the case with sub-Kolmogorov bubbles. Instead, it is caused by the nonlinear drag–velocity relationship, where velocity fluctuations lead to an increased average drag. For $Fr > 0.5$, we confirm the scaling $\tilde {w}_b / w_b \propto 1 / Fr$, as proposed previously by Ruth et al. (J. Fluid Mech., vol. 924, 2021, p. A2), over a wide range of bubble inertia and deformability.

Diffusion-driven flow is a boundary layer flow arising from the interplay of gravity and diffusion in density-stratified fluids when a gravitational field is non-parallel to an impermeable solid boundary. This study investigates diffusion-driven flow within a nonlinearly density-stratified fluid confined between two tilted parallel walls. We introduce an asymptotic expansion inspired by the centre manifold theory, where quantities are expanded in terms of derivatives of the cross-sectional averaged stratified scalar (such as salinity or temperature). This technique provides accurate approximations for velocity, density and pressure fields. Furthermore, we derive an evolution equation describing the cross-sectional averaged stratified scalar. This equation takes the form of the traditional diffusion equation but replaces the constant diffusion coefficient with a positive-definite function dependent on the solution's derivative. Numerical simulations validate the accuracy of our approximations. Our investigation of the effective equation reveals that the density profile depends on a non-dimensional parameter denoted as $\gamma$ representing the flow strength. In the large $\gamma$ limit, the system is approximated by a diffusion process with an augmented diffusion coefficient of $1+\cot ^{2}\theta$, where $\theta$ signifies the inclination angle of the channel domain. This parameter regime is where diffusion-driven flow exhibits its strongest mixing ability. Conversely, in the small $\gamma$ regime, the density field behaves like pure diffusion with distorted isopycnals. Lastly, we show that the classical thin film equation aligns with the results obtained using the proposed expansion in the small $\gamma$ regime but fails to accurately describe the dynamics of the density field for large $\gamma$.

We examine the mechanisms responsible for the onset of the three-dimensional mode B instability in the wake behind a circular cylinder. We show that it is possible to explicitly account for the stabilising effect of spanwise viscous diffusion and then demonstrate that the remaining mechanisms involved in this short-wavelength instability are preserved in the limit of zero wavelength. Using the resulting simplified equations, we show that perturbations in different fluid particles interact only through the in-plane viscous diffusion which turns out to have a destabilising effect. We also show that in the presence of viscous diffusion, the closed trajectories which had been conjectured to play a crucial role in the onset of the mode B instability are not actually a prerequisite for the growth of mode B type perturbations. We combine these observations to identify the three essential ingredients for the development of the mode B instability: (i) the amplification of perturbations in the braid regions due to the stretching mechanism; and the spreading of perturbations through (ii) viscous diffusion, and (iii) cross-flow advection which transports fluid between the two braid regions on either side of the cylinder. Finally, we develop a simple criterion that allows the prediction of the regions where three-dimensional short-wavelength perturbations are amplified by the stretching mechanism. The approach used in our study is general and has the potential to give insights into the onset of three-dimensionality via short-wavelength instabilities in other flows.

The nonlinear waves in a sheared liquid film on a horizontal plate at small Reynolds numbers are examined by theoretical and numerical approaches. The analysis employs the long-wave approximation along with finite difference schemes. The results show that the surface tension can suppress disturbances and prevent the occurrence of singularities. While the film flow is driven by the shear stress on the interface, its instability highly depends on the magnitude and direction of gravity. Specifically, when the direction of gravity is opposite to the wall-normal direction, perturbations are stabilized by gravity. In contrast, when these two directions are the same, the gravitational force is destabilizing, and stationary travelling waves can exist if a balance is reached between the effects of gravity and surface tension. For the steady solitary waves, there are quasi-periodic oscillations occurring between two stationary points, indicating the presence of heteroclinic trajectories. For periodic waves, the evolutions are sensitive to several parameters and initial disturbances, while one steady-state wave exhibits a sine function-like behaviour.

We studied the transport and deposition behaviour of point particles in Rayleigh–Bénard convection cells subjected to Couette-type wall shear. Direct numerical simulations (DNSs) are performed for Rayleigh number ($Ra$) in the range $10^{7} \leq Ra \leq ~10^9$ with a fixed Prandtl number $Pr = 0.71$, while the wall-shear Reynolds number ($Re_w$) is in the range $0 \leq Re_w \leq ~12\,000$. With the increase of $Re_w$, the large-scale rolls expanded horizontally, evolving into zonal flow in two-dimensional simulations or streamwise-oriented rolls in three-dimensional simulations. We observed that, for particles with a small Stokes number ($St$), they either circulated within the large-scale rolls when buoyancy dominated or drifted near the walls when shear dominated. For medium $St$ particles, pronounced spatial inhomogeneity and preferential concentration were observed regardless of the prevailing flow state. For large $St$ particles, the turbulent flow structure had a minor influence on the particles’ motion; although clustering still occurred, wall shear had a negligible influence compared with that for medium $St$ particles. We then presented the settling curves to quantify the particle deposition ratio on the walls. Our DNS results aligned well with previous theoretical predictions, which state that small $St$ particles settle with an exponential deposition ratio and large $St$ particles settle with a linear deposition ratio. For medium $St$ particles, where complex particle–turbulence interaction emerges, we developed a new model describing the settling process with an initial linear stage followed by a nonlinear stage. Unknown parameters in our model can be determined either by fitting the settling curves or using empirical relations. Compared with DNS results, our model also accurately predicts the average residence time across a wide range of $St$ for various $Re_w$.

Isolated-roughness-induced transitions controlled by local wall heating strips are studied via direct numerical simulation and BiGlobal linear stability analysis. The transition mechanisms are studied first with different wall temperatures. The separated shear layer–counter-rotating vortex system is found to be the main source for transitions. Symmetric and antisymmetric modes are observed in the wake, and the former is dominant. The local wall heating strip can delay the transition, and this effect is enhanced with higher heating temperature, wider strip and a combination of upstream and downstream control strips. The upstream strip lifts up the inlet flow and weakens the wake system in an indirect manner. The antisymmetric mode gradually vanishes, while the symmetric mode always exists but becomes weaker. The downstream strip exhibits a more effective transition delay by directly weakening the separated shear layer and vortex system in the wake. Vorticity transport analysis suggests that the downstream strip increases dissipation for streamwise vorticity and transfers it into wall-normal and spanwise vorticity. BiGlobal analyses indicate that the downstream strip shows less influence on the peak growth rate of the symmetric mode but significantly shrinks its unstable region. Analyses of the disturbance energy production indicate that the upstream strip wakens the wall-normal and spanwise shear at the same time, but the downstream strip mainly wakens the wall-normal one. More simulations are performed with different roughness heights, point-source disturbance and different roughness shapes. The results show that the current method remains effective enough in delaying transitions at a wide range of conditions.

The penetration of a spherical vortex into turbulence is studied theoretically and experimentally. The characteristics of the vortex are first analysed from an integral perspective that reconciles the far-field dipolar flow with the near-field source flow. The influence of entrainment on the vortex drag force is elucidated, extending the Maxworthy (J. Fluid Mech., vol. 81, 1977, pp. 465–495) model to account for turbulent entrainment into the vortex movement and vortex penetration into an evolving turbulent field. The physics are explored numerically using a spherical vortex (initial radius $R_0$, speed $U_{v0}$), characterised by a Reynolds number $Re_0(=2R_0U_{v0}/\nu$, where $\nu$ is the kinematic viscosity) of 2000, moving into decaying homogeneous turbulence (root-mean-square $u_0$, integral scale $L$) with turbulent intensity $I_t=u_0/U_{v0}$. When the turbulence is absent ($I_t=0$), a wake volume flux leads to a reduction of vortex impulse that causes the vortex to slow down. In the presence of turbulence ($I_t> 0$), the loss of vortical material is enhanced and the vortex speed decreases until it is comparable to the local turbulent intensity and quickly fragments, penetrating a distance that scales as $I_t^{-1}$. In the experimental study, a vortex ($Re_0\sim 1490\unicode{x2013}5660$) propagating into a statistically steady, spatially varying turbulent field ($I_{ve}=0.02$ to 0.98). The penetration distance is observed to scale with the inverse of the turbulent intensity. Incorporating the spatially and temporally varying turbulent fields into the integral model gives a good agreement with the predicted trend of the vortex penetration distance with turbulent intensity and insight into its dependence on the structure of the turbulence.

This study investigates the interaction between a freely rising, deformable bubble and a freely settling particle of the same size due to gravity. Initially, an in-line configuration is considered while varying the Bond, Galilei and Archimedes numbers. The study shows that as the bubble and particle approach each other, a liquid film forms between them that undergoes drainage. The formation of the liquid film leads to dissipation of kinetic energy, and for sufficiently large bubble velocities, particle flotation takes place. Increasing the Bond number causes the bubble to deform more severely, which may allow the particle to pass through the bubble as it ruptures. This work also considers an offset configuration, which shows that the bubble slides away from the particle, affecting its settling trajectory.

Transonic aeroelasticity remains a significant challenge in aerospace. The coupling mechanism of aeroelastic problems involving the coexistence of fluid modes and multiple structural modes still needs further investigation. For this purpose, we analysed the dynamic characteristic of a two-degree-of-freedom (2DOF) NACA0012 airfoil in pre-buffet flow. First, we constructed an aeroelastic reduced-order model, which can represent near-unstable transonic flow using the dominant fluid mode. Then, the flutter mechanism was investigated by studying the main eigenvalues of the model that vary with the natural pitching frequency. The results revealed that the existence of the fluid mode transitions the transonic flutter type from coupled-mode flutter to single-DOF (SDOF) flutter, which leads to a reduction in the flutter boundary. Under the effect of the fluid mode, the system produces six aeroelastic phenomena at different structural natural frequencies, including SDOF heaving/pitching flutter, heaving/pitching instability within coupled-mode flutter, forced vibration and stable state. Moreover, we identified two types of SDOF flutter in the 2DOF system. The first type corresponds to the traditional SDOF flutter, where the coupling of other modes has a small impact on the system's stability in most cases. However, within specific ranges of natural frequencies, this type of SDOF flutter may disappear due to coupling with other modes. The second type of SDOF flutter is characterized by strong coupling dominated by the unstable mode. It arises from the interaction among the flow, heaving and pitching modes, and does not manifest in the absence of any of these modes.

We investigate the heterogeneous cavitation phenomenon in water when a spherical surface is abruptly separated from a nearby flat substrate, at a distance of approximately 10 nm. By tracking the surface separation using Newton ring positions, and capturing the bubble evolution with a high-speed camera on a microscope, we compare our experimental findings with hydrodynamic predictions at low Reynolds numbers. Upon upward movement of the spherical surface, the resulting bubble develops branched fingers through the Saffman–Taylor instability. Simultaneously, negative liquid pressures in the range $\sim$10 atm are observed. These large tension values occasionally lead to secondary nucleation events. The bubble sizes satisfy a predicted Family–Vicsek scaling law where the bubble area is proportional to the inverse bubble lifetime. The fact that creeping flow cavitation bubbles are more short-lived the larger they are separates them from bubbles that are governed by inertial dynamics.