In most cases, the dominant mechanism of energy dissipation for a bubble in volume oscillations is the thermal energy exchanged with the liquid. The process is subtle and its precise description a matter of some complexity. These features have prevented its ready incorporation in many applications, which forcedly have to rely on the rather inaccurate polytropic pressure–volume relation. This paper develops two approximate models of the thermal interaction, formulated in terms of ordinary differential equations, which can be readily added to standard Rayleigh–Plesset-type formulations at a modest computational cost.

The aim of this paper is to make the formation of liquid plugs as difficult as possible in liquid partially filling a horizontal rectangular tube in a downward gravity field by setting the walls to have differing contact angles. Manning et al.'s method (J. Fluid Mech., vol. 682, 2011, pp. 397–414), extended from Concus–Finn theory, is applied to the existence of capillary plugs in rectangular tubes. The critical Bond numbers ($B_c$) determining the existence of capillary plugs in a rectangular tube are studied for different settings of the non-uniform contact angles, and the influence of the aspect ratio (defined as the width-to-height ratio) of the rectangular cross-section on $B_c$ is examined. Compared to the maximum and minimum of $B_c$ reached for uniform contact angles, the maximum of $B_c$ is higher, which is attained for the bottom contact angle $\gamma _2=135^{\circ }$, the top contact angle $\gamma _4=45^{\circ }$, and the side contact angles $\gamma _1=\gamma _3=90^{\circ }$; while the minimum is considerably lowered to zero, which is reached for $\gamma _1=\gamma _2=45^{\circ }$ and $\gamma _3=\gamma _4=135^{\circ }$. The aspect ratio of the rectangle has no influence on the maximum and minimum $B_c$ for a tube with walls of differing contact angles. There is only one non-occluded liquid topology in a square, while two topologies may occur in a rectangle with aspect ratio 2, and the transition between the two topologies is accompanied by a kink of the curve of $B_c$. Optimization of the non-uniform contact angles can facilitate or effectively block the capillary plugs in rectangular tubes regardless of the aspect ratios.

Fluid deformable surfaces show a solid–fluid duality which establishes a tight interplay between tangential flow and surface deformation. We derive the governing equations as a thin film limit and provide a general numerical approach for their solution. The simulation results demonstrate the rich dynamics resulting from this interplay, where, in the presence of curvature, any shape change is accompanied by a tangential flow and, vice versa, the surface deforms due to tangential flow. However, they also show that the only possible stable stationary state in the considered setting is a sphere with zero velocity.

In the preparation of café latte, spectacular layer formation can occur between the espresso shot in a glass of milk and the milk itself. Xue et al. (Nat. Commun., vol. 8, 2017, pp. 1–6) showed that the injection velocity of espresso determines the depth of coffee–milk mixture. After a while, when a stable stratification forms in the mixture, the layering process can be modelled as a double diffusive convection system with a stably stratified coffee–milk mixture cooled from the side. More specifically, we perform (two-dimensional) direct numerical simulations of laterally cooled double diffusive convection for a wide parameter range, where the convective flow is driven by a lateral temperature gradient while stabilized by a vertical concentration gradient. Depending on the strength of stabilization as compared to the thermal driving, the system exhibits different flow regimes. When the thermal driving force dominates over the stabilizing force, the flow behaves like vertical convection in which a large-scale circulation develops. However, with increasing strength of the stabilizing force, a meta-stable layered regime emerges. Initially, several vertically-stacked convection rolls develop, and these well-mixed layers are separated by sharp interfaces with large concentration gradients. The initial thickness of these emerging layers can be estimated by balancing the work exerted by thermal driving and the required potential energy to bring fluid out of its equilibrium position in the stably stratified fluid. In the layered regime, we further observe successive layer merging, and eventually only a single convection roll remains. We elucidate the following merging mechanism: as weakened circulation leads to accumulation of hot fluid adjacent to the hot sidewall, larger buoyancy forces associated with hotter fluid eventually break the layer interface. Then two layers merge into a larger layer, and circulation establishes again within the merged structure.

A roughness scaling behaviour is tested by performing the direct numerical simulation (DNS) of a turbulent channel flow over three-dimensional sinusoidal rough walls. By systematically varying the roughness height ${{k}^{+}}$ and the roughness steepness S, the results for three groups of cases are considered and compared with those for flat-wall turbulence. The results show that the mean velocity and Reynolds stresses are highly dependent on both ${{k}^{+}}$ and S. To describe these specific relationships, we define a coupling scale ${{k}^{+}} S$. With this coupling scale, all the simulated data for the roughness function (${\rm \Delta} {{U}^{+}}$), the ratio of the pressure drag to the total wall resistance (${{\gamma }_{p}}$), the normalized bulk mean velocity ($U_{b}^{+}$) and the peak of the streamwise turbulent velocity fluctuations ($\overline {u_{p}^{\prime +}}$) collapse onto single curves, which shows that there is a strong direct correlation between them, i.e. ${\rm \Delta} U^{+}, \gamma _{p}, U_{b}^{+}, \overline {u_{p}^{\prime +}} \propto f(k^{+} S)$. Furthermore, a model for the prediction of wall resistance based on the roughness function can be established by defining a drag increasing ratio (DI). Accordingly, the wall resistance coefficient ${{C}_{f}}$ can be estimated directly from ${{k}^{+}}S$ of a given rough surface. These results suggest that this coupling scale provides a useful alternative to the equivalent sand grain roughness ${{k}_{s}}$.

Linearisation of the Navier–Stokes equations about the mean of a turbulent flow forms the foundation of popular models for energy amplification and coherent structures, including resolvent analysis. While the Navier–Stokes equations can be equivalently written using many different sets of dependent variables, we show that the properties of the linear operator obtained via linearisation about the mean depend on the variables in which the equations are written prior to linearisation, and can be modified under nonlinear transformation of variables. For example, we show that using primitive and conservative variables leads to differences in the singular values and modes of the resolvent operator for turbulent jets, and that the differences become more severe as variable-density effects increase. This lack of uniqueness of mean-flow-based linear analysis provides new opportunities for optimising models by specific choice of variables while also highlighting the importance of carefully accounting for the nonlinear terms that act as a forcing on the resolvent operator.

Reconnection plays a significant role in the dynamics of plasmas, polymers and macromolecules, as well as in numerous laminar and turbulent flow phenomena in both classical and quantum fluids. Extensive studies in quantum vortex reconnection show that the minimum separation distance $\delta$ between interacting vortices follows a $\delta(t) \sim t^{1/2}$ scaling. Due to the complex nature of the dynamics (e.g. the formation of bridges and threads as well as successive reconnections and avalanche), such scaling has never been reported for (classical) viscous vortex reconnection. Using direct numerical simulation of the Navier–Stokes equations, we study viscous reconnection of slender vortices, whose core size is much smaller than the radius of the vortex curvature. For separations that are large compared to the vortex core size, we discover that $\delta (t)$ between the two interacting viscous vortices surprisingly also follows the 1/2-power scaling for both pre- and post-reconnection events. The prefactors in this 1/2-power law are found to depend not only on the initial configuration but also on the vortex Reynolds number (or viscosity). Our finding in viscous reconnection, complementing numerous works on quantum vortex reconnection, suggests that there is indeed a universal route for reconnection – an essential result for understanding the various facets of the vortex reconnection phenomena and their potential modelling, as well as possibly explaining turbulence cascade physics.

We explore the near-resonant interaction of inertial waves with geostrophic modes in rotating fluids via numerical and theoretical analysis. When a single inertial wave is imposed, we find that some geostrophic modes are unstable above a threshold value of the Rossby number $kRo$ based on the wavenumber and wave amplitude. We show this instability to be caused by triadic interaction involving two inertial waves and a geostrophic mode such that the sum of their eigenfrequencies is non-zero. We derive theoretical scalings for the growth rate of this near-resonant instability. The growth rate scaled by the global rotation rate is proportional to $(kRo)^2$ at low $kRo$ and transitions to a $kRo$ scaling for larger $kRo$. These scalings are in excellent agreement with direct numerical simulations. This instability could explain recent experimental observations of geostrophic instability driven by waves.

The dynamics of a stably and thermally stratified fluid-filled cavity harmonically forced in the vertical direction, resulting in a periodic gravity modulation, is studied numerically. Prior simulations in a two-dimensional cavity showed a myriad of complex dynamic behaviours near the onset of instabilities, and here we address the extent to which these persist in three dimensions. Focusing on a parameter regime where the primary subharmonic mode is resonantly driven, we demonstrate comprehensive qualitative agreement between the dynamics in two and three dimensions; the quantitative difference is due to the larger forcing amplitudes needed in three dimensions to overcome the additional viscous damping from the spanwise walls. Using a small detuning of the forcing frequency, together with a relatively large forcing amplitude, leads to a wave-breaking regime where the qualitative agreement between two and three dimensions breaks down.

We report on a numerical study of turbulent convection driven by a combination of internal heat sources and sinks. Motivated by a recent experimental realisation (Lepot etal., Proc. Natl Acad. Sci. USA, vol. 115 (36), 2018, pp. 8937–8941), we focus on the situation where the cooling is uniform, while the internal heating is localised near the bottom boundary, over approximately one tenth of the domain height. We obtain scaling laws ${Nu} \sim {Ra} ^{\gamma } {Pr}^{\chi }$ for the heat transfer as measured by the Nusselt number ${Nu}$ expressed as a function of the Rayleigh number ${Ra}$ and the Prandtl number ${Pr}$. After confirming the experimental value $\gamma \approx 1/2$ for the dependence on ${Ra}$, we identify several regimes of dependence on ${Pr}$. For a stress-free bottom surface and within a range as broad as ${Pr} \in [0.003, 10]$, we observe the exponent $\chi \approx 1/2$, in agreement with Spiegel's mixing-length theory. For a no-slip bottom surface we observe a transition from $\chi \approx 1/2$ for ${Pr} \leq 0.04$ to $\chi \approx 1/6$ for ${Pr} \geq 0.04$, in agreement with scaling predictions by Bouillaut etal. (J. Fluid Mech. vol. 861, 2019, R5). The latter scaling regime stems from heat accumulation in the stagnant layer adjacent to a no-slip bottom boundary, which we characterise by comparing the local contributions of diffusive and convective thermal fluxes.

In this paper, we consider two-dimensional steady free-surface flows where gravity is ignored, but the effects of surface tension are included. It is found that the existence of an additional solid boundary can allow for previously unseen limiting configurations as the surface tension tends to infinity. The free surface of these new solutions is formed of straight lines, arcs of circles and a point where the flow turns at a wall. These limiting configurations form endpoints of solution branches of capillary free-surface flows. Other endpoints of these branches include the surface tension free (i.e. free streamline) solution, and a solution whose free surface is composed simply of a straight line. The model we explore is flow incoming along a channel of constant width. One of the walls terminates, where the fluid is forcibly separated from the wall and a free boundary is formed. The other wall meets a second straight boundary with interior angle $\beta$. Far downstream the solution approaches a uniform stream. Making use of Cauchy's integral formula, the unknowns are expressed in terms of values on the boundary. The integral equations are then solved numerically. The solution space relative to the parameter values of the model is discussed.

We study the longitudinal spreading of a passive tracer by a two-dimensional pressure-driven flow through a composite layer of porous rock which is bounded above and below by impermeable seal rock. We focus on the flow across the interface between two neighbouring zones of the rock. First, we show that, with isotropic permeability, if the interface between the two zones is tilted relative to the boundaries, then this results in a difference in travel times across the formation which in turns leads to a net shear flow. We explore the strength of this shear as a function of (a) the permeability ratio across the interface, and (b) the interface angle. Second, we show that if one zone of the rock is cross-bedded, then with uniform flow, the pressure gradient is directed at an angle to the boundary. As a result, there is a transition zone across the interface, which again leads to a net shear, even if the interface is normal to the boundaries of the layer. We explore the competition between these effects, showing how they may combine constructively to produce a larger shear, or may negate one another, reducing or reversing the sign of the shear, depending on the angle of the interface, the degree of anisotropy and the change in effective downstream permeability across the interface. We discuss some of the implications of this shear for modelling flow in such composite rocks.

The flow in a channel having walls with periodic undulations of small amplitude $\epsilon$ in the streamwise direction is considered as a model for wall roughness. It is shown that the undulations act as a catalyst to allow a new instability related to vortex–wave interactions to grow. The roughness couples a wave disturbance with a roll–streak flow and it is shown that channel flows, both wall and pressure gradient driven, are unstable when the Reynolds number exceeds a critical value proportional to ${\epsilon ^{-({3}/{2})} [\vert {\log \epsilon }\vert ]^{-({3}/{4})}}$, the constant of proportionality depending on the wall wavelengths and amplitudes. The roughness is an integral part of the instability mechanism and not simply the seed for an existing flow instability as in receptivity theory. The mechanism involves an interaction of the rolls, streaks and waves very similar to that in vortex–wave interaction theory but now facilitated by the wall roughness. Surprisingly, the subtle interaction between waves, rolls, streaks and the walls can be solved in closed form, and an explicit form for the neutral configuration is found. The theoretical predictions are in good agreement with numerical investigations of similar problems and are applicable to a wide range of shear flows.

This paper addresses the pulsating motion of cerebrospinal fluid in the aqueduct of Sylvius, a slender canal connecting the third and fourth ventricles of the brain. Specific attention is given to the relation between the instantaneous values of the flow rate and the interventricular pressure difference, needed in clinical applications to enable indirect evaluations of the latter from direct magnetic resonance measurements of the former. An order of magnitude analysis accounting for the slenderness of the canal is used in simplifying the flow description. The boundary layer approximation is found to be applicable in the slender canal, where the oscillating flow is characterized by stroke lengths comparable to the canal length and periods comparable to the transverse diffusion time. By way of contrast, the flow in the non-slender opening regions connecting the aqueduct with the two ventricles is found to be inviscid and quasi-steady in the first approximation. The resulting simplified description is validated by comparison with results of direct numerical simulations. The model is used to investigate the relation between the interventricular pressure and the stroke length, in parametric ranges of interest in clinical applications.

State estimation is key to both analysing physical mechanisms and enabling real-time control of fluid flows. A common estimation approach is to relate sensor measurements to a reduced state governed by a reduced-order model (ROM). (When desired, the full state can be recovered via the ROM.) Current methods in this category nearly always use a linear model to relate the sensor data to the reduced state, which often leads to restrictions on sensor locations and has inherent limitations in representing the generally nonlinear relationship between the measurements and reduced state. We propose an alternative methodology whereby a neural network architecture is used to learn this nonlinear relationship. A neural network is a natural choice for this estimation problem, as a physical interpretation of the reduced state–sensor measurement relationship is rarely obvious. The proposed estimation framework is agnostic to the ROM employed, and can be incorporated into any choice of ROMs derived on a linear subspace (e.g. proper orthogonal decomposition) or a nonlinear manifold. The proposed approach is demonstrated on a two-dimensional model problem of separated flow around a flat plate, and is found to outperform common linear estimation alternatives.

We study the impact between a plate and a drop of non-colloidal solid particles suspended in a Newtonian liquid, paying specific attention to the case when the particle volume fraction, $\unicode[STIX]{x1D719}$, is close to – or even exceeds – the critical volume fraction, $\unicode[STIX]{x1D719}_{c}$, at which the steady effective viscosity of the suspension diverges. We use a specific concentration protocol together with an accurate determination of $\unicode[STIX]{x1D719}$ for each drop, and we measure the deformation $\unicode[STIX]{x1D6FD}$ for different liquid viscosities, impact velocities and particle sizes. At low volume fractions, $\unicode[STIX]{x1D6FD}$ is found to follow closely an effective Newtonian behaviour, which we determine by documenting the low-deformation limit for a highly viscous Newtonian drop and characterizing the effective shear viscosity of our suspensions. By contrast, whereas the effective Newtonian approach predicts that $\unicode[STIX]{x1D6FD}$ vanishes at $\unicode[STIX]{x1D719}_{c}$, a finite deformation is observed for $\unicode[STIX]{x1D719}>\unicode[STIX]{x1D719}_{c}$. This finite deformation remains controlled by the suspending liquid viscosity and increases with increasing particle size, which suggests that the dilatancy of the particle phase is a key factor in the dissipation process close to and above $\unicode[STIX]{x1D719}_{c}$.

We derive weakly dispersive Boussinesq equations using a $\unicode[STIX]{x1D70E}$ coordinate for the vertical direction, employing a series expansion in powers of $\unicode[STIX]{x1D70E}$. We restrict attention initially to the case of constant still-water depth $h$ in order to simplify subsequent analysis, and consider equations based on expansions about the bottom elevation $\unicode[STIX]{x1D70E}=0$, and then about a reference elevation $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x1D6FC}}$ in order to improve linear dispersion properties. We use a perturbation analysis, suggested recently by Madsen & Fuhrman (J. Fluid Mech., vol. 889, 2020, A38), to show that the resulting models are not subject to the trough instability studied there. A similar analysis is performed to develop a model for interfacial waves in a two-layer fluid, with comparable results. We argue, by extension, that a necessary condition for eliminating trough instabilities is that the model’s nonlinear dispersive terms should not contain still-water depth $h$ and surface displacement $\unicode[STIX]{x1D702}$ separately.

We present an exact solution for periodic travelling waves in two-dimensional, infinitely deep and constant vorticity flows, in the absence of the effects of gravity or surface tension. The shape of the free surface is the same as for Crapper’s celebrated capillary waves in an irrotational flow, but the flow beneath the wave, which is also explicit, is completely different. This confirms a conjecture made by Dyachenko & Hur (J. Fluid Mech., vol. 878, 2019b, pp. 502–521; Stud. Appl. Maths, vol. 142 (2), 2019c, pp. 162–189) and Hur & Vanden-Broeck (Eur. J. Mech. (B/Fluids), 2020, to appear), based on numerical and asymptotic evidence.

Internal solitary and solitary-like waves (ISW) play an important role in mixing and sediment resuspension in naturally occurring stratified fluids, primarily through various instabilities and wave-breaking mechanisms. When shoaling into shallow waters, waves of depression may either fission into a packet of waves of elevation over mild slopes or break over steep slopes. The fissioning process is generally considered a less efficient transport and resuspension mechanism, compared to wave breaking, since very little turbulent mixing or energy dissipation occurs during this process. In the present work, however, we found that this is not always the case, at least in the particular context of ISW boundary-layer interaction. Using high-resolution numerical simulations performed in a domain representing a tilted laboratory tank, we found that boundary-layer instability in the form of a separation bubble consistently occurs during the fissioning process. The separation bubble is generated beneath the wave of elevation that emerges from the fissioning process, and is vitally influenced by currents induced by the leading wave of depression. As the waves shoal further, the growth and breakdown of the separation bubble leads to significant cross-boundary-layer transport. The results suggest that the fissioning process, which occurs over a considerable geographical region in the ocean, can be as efficient as wave breaking when it comes to cross-boundary-layer transport.

Experiments on the receptivity of two-dimensional boundary layers to acoustic disturbances from two-dimensional roughness strips were performed in a low-turbulence wind tunnel on a flat plate model. The free stream was subjected to a plane acoustic wave so that a Stokes layer (SL) was created on the plate, thus generating a Tollmien–Schlichting (T–S) wave through the receptivity process. An improved technique to measure the T–S component is described based on a retracting two-dimensional roughness, which allowed for phase-locked measurements at the acoustic wave frequency to be made. This improved technique enables both protuberances and cavities to be explored in the range $30~\unicode[STIX]{x03BC}\text{m}<|h|<750~\unicode[STIX]{x03BC}\text{m}$ (equivalent to $0.025<|h|/\unicode[STIX]{x1D6FF}_{B}^{\ast }<0.630$ in relative roughness height to the local unperturbed Blasius boundary layer displacement thickness). These depths are designed to cover both the predicted linear and nonlinear response of the T–S excitation. Experimentally, cavities had not previously been explored. Results show that a linear regime is identifiable for both positive and negative roughness heights up to ${\approx}150~\unicode[STIX]{x03BC}\text{m}$ ($|h|/\unicode[STIX]{x1D6FF}_{B}^{\ast }\approx 0.126$). The departure from the linear behaviour is, however, dependent on the geometry of the surface imperfection. For cavities of significant depth, the nonlinear behaviour is found to be milder than in the case of protuberances – this is attributed to the flow physics in the near field of the surface features. Nonetheless, results for positive heights agree well with previous theoretical work which predicted a linear disturbance response for small-height perturbations.