A cavitation bubble expanding and collapsing near a rigid boundary develops a directed jet flow towards the boundary. In the case of a perforated plate, some of the jet flow passes through the plate and thus the bubble acts as a pump transporting liquid from one side of the plate to the opposite side. The transport is rather complex, is time dependent and varies with the geometric parameters of the bubble and the connecting channel. Therefore, we first model the transport of liquid through a perforated rigid plate for a large range of parameters and then compare some regimes with experiments using single laser-induced bubbles. The simulations are based on a Volume-of-Fluid solver in OpenFOAM and account for surface tension, compressibility and viscosity. The resulting flux and generated velocity in the channel obtained in the simulations are discussed with regards to the dependence of the channel geometry, liquid viscosity and stand-off distance of the bubble to the plate. In general, high flow rates are achieved for long cylindrical channels that have a similar width as the jet produced by the collapsing bubble. At low stand-off distances combined with thick plates, an annular inflow creates a fast and thin jet, also called needle jet, which is approximately a magnitude faster and significantly thinner than the usually encountered microjet. In contrast, for thin plates and small stand-off distances, liquid is pumped in the opposite direction via a reverse jet.

Inertial transitions in a suspension Taylor–Couette flow are explored for the first time via experiments in the counter-rotation regime (i.e. the rotation ratio $\varOmega =\omega _o/\omega _i<0$, where $\omega _i$ and $\omega _o$ are the angular speeds of the inner and outer cylinders, respectively) up to a particle volume fraction $\phi \leq 0.2$. The primary bifurcation from the circular Couette flow (CCF) is found to yield patterns similar to those in a particle-free Newtonian fluid, and is supercritical at $|\varOmega |\leq 0.5$ but becomes hysteretic at large $|\varOmega |=1$. It is shown that the states with different numbers of vortices can coexist over a large range of shear Reynolds number $\mbox {Re}_s(\phi )$, confirming multi-stable behaviour of the primary and higher-order states for any particle loading at $|\varOmega |\leq 0.5$. A quasi-periodic state characterized by two incommensurate frequencies, namely, the modulated wavy vortices (MWV), is found at $\varOmega =-0.5$ as a tertiary bifurcation from the wavy Taylor vortices (WTV), with the stationary Taylor vortex flow (TVF) being the primary bifurcating state from CCF at $\phi \geq 0$. A novel sequence of transitions ${\rm TVF}\to {\rm MWV}_1\to {\rm WTV}\to {\rm MWV}$, with another variant of modulated vortices (MWV$_1$) appearing directly from TVF as a secondary bifurcation, may also occur even for the particle-free ($\phi =0$) case; the coexistence/non-uniqueness of WTV and MWV states is demonstrated over a range of $\mbox {Re}_s$ values spanning the secondary and tertiary bifurcation loci. At $\varOmega =-1$, the primary bifurcation yields an oscillatory state (spiral/helical vortex flow) that gives birth to another oscillatory state (interpenetrating spiral vortices) and a non-periodic state (non-propagating interpenetrating spirals, NIS) as secondary and tertiary bifurcations, respectively, with NIS being characterized by the absence of frequency peaks in the power spectrum of the scattered light intensity. For all transitions at $\varOmega < 0$, the critical values of $\mbox {Re}_s^c(\phi )$ decrease with increasing $\phi$, with more destabilizing effects of particles being found at larger $|\varOmega |$. The effect of particle loading is found to (i) decrease the amplitude and (ii) increase the wavelength of wavy vortices, with the latter seeming to be responsible for the decreased propagation frequencies of azimuthal waves with increasing $\phi$. The normalized rotation frequency of spiral vortices also decreases with increasing $\mbox {Re}_s$ and $\phi$, and a scaling relation in terms of the relative viscosity is found to collapse the frequency data for all $\phi$.

Using fluid–structure interaction computational modelling, the hydrodynamic performance of bio-inspired elastic propulsors with tapered thickness that oscillate in an incompressible Newtonian fluid at Reynolds number $Re = 2000$ is investigated. The thickness tapering leads to an acoustic black hole effect at the trailing edge of the propulsor that slows down and attenuates flexural waves, thereby minimizing the flexural wave reflection and enhancing travelling wave propulsion. The simulations reveal that, by tuning the propulsor thickness profile modulating the acoustic black hole effect, the tapered propulsors can be designed to vastly outperform the uniformly thick propulsors in terms of the hydrodynamic efficiency and thrust, especially for the post-resonance frequencies. The enhanced hydrodynamic performance is directly linked to the ability of the tapered propulsors to generate travelling waves with a large amplitude displacement at the trailing edge. The results have implications for the development of highly efficient bio-mimetic robotic swimmers and, more generally, the better understanding of the undulatory aquatic locomotion.

We present a self-propelled axisymmetric low-Reynolds-number swimmer (force- and torque-free) that, while unable to swim in a Newtonian fluid, propels itself in a non-Newtonian fluid as a result of fluid elasticity. The propulsion gait is pure ‘swirling’ and is demonstrated using a robotic swimmer, comprised of a ‘head’ sphere and a ‘tail’ sphere, whose swimming speed is shown to demonstrate good agreement with a microhydrodynamic asymptotic theory and numerical simulations. Schlieren imaging demonstrates that propulsion of the swimmer is driven by a strong viscoelastic jet at the tail, which develops due to the fore–aft asymmetry of the swimmer. Optimized cylindrical and conic tail geometries are shown to double the propulsive signal, relative to the optimal spherical tail. We show that we can use observations of this robot to infer rheological properties of the surrounding fluid. We measure the primary normal stress coefficient $\varPsi _1$ at shear rates ${<}1\,{\rm Hz}$, and show reasonable agreement with extrapolated bench-top measurements ($0.8\unicode{x2013}1.2\,{\rm Pa}\,{\rm s}^2$ difference). We also derive how the head rotation may be used to measure the second normal stress coefficient, $\varPsi _2$. The study demonstrates the exciting potential for a ‘swimming rheometer’, examining the intersection between swimming and sensing in complex fluids.

This work explains a scaling law of the first Landau coefficient of the derived Ginzburg–Landau equation in the weakly nonlinear analysis of axisymmetric viscoelastic pipe flows in the large-Weissenberg-number ($Wi$) limit, recently reported in Wan et al. (J. Fluid Mech., vol. 929, 2021, A16). Using an asymptotic method, we derive a reduced system, which captures the characteristics of the linear centre-mode instability near the critical condition in the large-$Wi$ limit. Based on the reduced system we then conduct a weakly nonlinear analysis using a multiple-scale expansion method, which readily explains the aforementioned scaling law of the Landau coefficient and some other scaling laws. Particularly, the equilibrium amplitude of disturbance near linear critical conditions is found to scale as $Wi^{-1/2}$, which may be of interest to experimentalists. The current analysis reduces the numbers of parameters and unknowns and exemplifies an approach to studying the viscoelastic flow at large $Wi$, which could shed new light on the understanding of its nonlinear dynamics.

One mechanism for airway closure in the lung is the surface-tension-driven instability of the mucus layer which lines the airway wall. We study the instability of an axisymmetric layer of viscoplastic Bingham liquid coating the interior of a rigid tube, which is a simple model for an airway that takes into account the yield stress of mucus. An evolution equation for the thickness of the liquid layer is derived using long-wave theory, from which we also derive a simpler thin-film evolution equation. In the thin-film case we show that two branches of marginally yielded static solutions of the evolution equation can be used to both predict the size of the initial perturbation required to trigger instability and quantify how increasing the capillary Bingham number (a parameter measuring yield stress relative to surface tension) reduces the final deformation of the layer. Using numerical solutions of the long-wave evolution equation, we quantify how the critical layer thickness required to form a liquid plug in the tube increases as the capillary Bingham number is increased. We discuss the significance of these findings for modelling airway closure in obstructive conditions such as cystic fibrosis, where the mucus layer is often thicker and has a higher yield stress.

We study drop impact for the case where the impacted surface is cooled below the freezing temperature of the liquid droplet. The freezing is found to affect the spreading dynamics of the impacting drops and, thus, the degree of surface coverage. The cooling of the surface leads to the arrest of the three-phase contact line, impeding droplet spreading and, thus, drastically reducing the maximum spreading diameter. Besides the surface temperature, the impact speed is also an important parameter: the higher the impact speed, the more the droplet spreads before arrest. Based on experimental observations of droplet impacts using two different liquids and two different substrates, we show using a combination of experiments and a one-dimensional freezing model, that droplet arrest occurs when a solid layer of the liquid forms on the substrate: droplet arrest occurs when this solid layer reaches a well-defined critical thickness. We then devise a simple model that efficiently predicts the maximum spreading diameter of droplets impinging, at different velocities, and freezing onto surfaces maintained at different temperatures below the liquid freezing point.

We present a family of exact equilibrium solutions of the Euler equations consisting of a single helical vortex with an axial flow component along the vortex core. Relations between vorticity, velocity and streamfunction are analytically derived in the inviscid framework and constitute a generalization of relations valid for vortex rings (Batchelor, 1967 An Introduction to Fluid Dynamics. Cambridge University Press). Through Navier–Stokes simulations, it is shown that these relations hold for quasi-steady viscous solutions and become independent of the Reynolds number when sufficiently large. We also elaborate a procedure which generates a quasi-equilibrium with prescribed characteristics (circulation, helix radius, helical pitch, vortex core size, swirl level) and compare the obtained state with the results of an asymptotic theory. Finally, we illustrate how a strong axial flow jeopardizes such an evolution towards a quasi-equilibrium.

A drag correlation is established for laminar particle-laden flows, based on data from the interfaced-resolved direct numerical simulations (IR-DNS) of particle sedimentation in a periodic domain at density ratio ranging from 2 to 1000, particle concentration ranging from 0.59 % to 14.16 %, and particle Reynolds number below 132. Our drag decreases slightly with increasing density ratio when the other parameters are fixed. The drag correlation is then corrected to account for the turbulence effect by introducing the relative turbulent kinetic energy, from the IR-DNS data of the upward turbulent channel flows laden with the particles larger than the Kolmogorov length scale at relatively low particle volume fractions. A drift velocity model is developed to obtain the effective slip velocity from the interphase mean velocity difference for the vertical turbulent channel flow by considering the effects of particle inertia, particle concentration distribution and large-scale streamwise vortices.

Physical representation of coherent motions in wall-bounded turbulence satisfying Townsend's attached-eddy hypothesis (AEH) is still an open question to be debated. Here, we apply proper orthogonal decomposition (POD) to instantaneous snapshots of $u$ component velocity fields in the streamwise–wall-normal plane of three turbulent boundary layers, which cover a frictional Reynolds number ($Re$) of $Re_\tau \approx 1000{\sim} 4000$ and were either measured by large-field-of-view planar particle image velocimetry or calculated by direct numerical simulation. The $u$ component POD modes characterized by streamwise recurring large-scale motions are decomposed into wall-attached and wall-detached parts depending on the wall-attachment condition. The former, i.e. wall-attached POD eddies, are believed to be the prime statistical representations of attached eddies. The reason lies in three aspects. Firstly, wall-attached POD eddies are geometrically self-similar and statistically independent. Secondly, the wall-normal variation of the variance of reduced-order streamwise velocity components $\tilde {u}_{AE}$, which is only contributed by wall-attached POD eddies, presents logarithmic decay in the inertial flow region. Thirdly, analysis on the scaling behaviour of the high-even-order moment, moment generating function, as well as longitudinal two-point structure function, all indicate a quasi-Gaussian behaviour of $\tilde {u}_{AE}$. None of these AEH-predicted behaviours are evident when the contribution of superstructure-like POD eddies are considered. Furthermore, wall-detached POD eddies are found to also present quasi-Gaussian behaviour. They might be treated as passive fragments of wall-attached ones.

Stationary longitudinal vortical rolls emerge in katabatic and anabatic Prandtl slope flows at shallow slopes as a result of an instability when the imposed surface buoyancy flux relative to the background stratification is sufficiently large. Here, we identify the self-pairing of these longitudinal rolls as a unique flow structure. The topology of the counter-rotating vortex pair bears a striking resemblance to speaker-wires and their interaction with each other is a precursor to further destabilization and breakdown of the flow field into smaller structures. On its own, a speaker-wire vortex retains its unique topology without any vortex reconnection or breakup. For a fixed slope angle $\alpha =3^{\circ }$ and at a constant Prandtl number, we analyse the saturated state of speaker-wire vortices and perform a bi-global linear stability analysis based on their stationary state. We establish the existence of both fundamental and subharmonic secondary instabilities depending on the circulation and transverse wavelength of the base state of speaker-wire vortices. The dominance of subharmonic modes relative to the fundamental mode helps to explain the relative stability of a single vortex pair compared to the vortex dynamics in the presence of two or an even number of pairs. These instability modes are essential for the bending and merging of multiple speaker-wire vortices, which break up and lead to more dynamically unstable states, eventually paving the way for transition towards turbulence. This process is demonstrated via three-dimensional flow simulations with which we are able to track the nonlinear temporal evolution of these instabilities.

The study of thermal convection in porous media is of both fundamental and practical interest. Typically, numerical studies have relied on the volume-averaged Darcy–Oberbeck–Boussinesq (DOB) equations, where convection dynamics are assumed to be controlled solely by the Rayleigh number (Ra). Nusselt numbers (Nu) from these models predict Nu–Ra scaling exponents of 0.9–0.95. However, experiments and direct numerical simulations (DNS) have suggested scaling exponents as low as 0.319. Recent findings for solutal convection between DNS and DOB models have demonstrated that the ‘pore-scale parameters’ not captured by the DOB equations greatly influence convection. Thermal convection also has the additional complication of different thermal transport properties (e.g. solid-to-fluid thermal conductivity ratio ks/kf and heat capacity ratio σ) in different phases. Thus, in this work we compare results for thermal convection from the DNS and DOB equations. On the effects of pore size, DNS results show that Nu increases as pore size decreases. Mega-plumes are also found to be more frequent and smaller for reduced pore sizes. On the effects of conjugate heat transfer, two groups of cases (Group 1 with varying ks/kf at σ = 1 and Group 2 with varying σ at ks/kf = 1) are examined to compare the Nu–Ra relations at different porosity (ϕ) and ks/kf and σ values. Furthermore, we report that the boundary layer thickness is determined by the pore size in DNS results, while by both the Rayleigh number and the effective heat capacity ratio, $\bar{\phi } = \phi + (1 - \phi )\sigma$, in the DOB model.

A droplet under a thermal gradient is known to migrate in a preferential direction, as governed by the variation of its interfacial tension with temperature. Contradicting the outcome of reported asymptotic analysis, here we show that the calculation of the droplet migration by considering the variation of interfacial tension with the imposed thermal field alone may be fundamentally incorrect. This error is attributed to the dynamically evolving interfacial temperature field due to a two-way coupling between the thermal field and the flow field, mediated by the droplet deformation and thermal diffusion. By directly capturing an inherent nonlinear coupling between the thermal field and the flow field using explicit interface tracking in a three-dimensional space, our boundary integral based analysis reveals that a linearly decreasing temperature profile imposed along the direction of a plane Poiseuille flow enhances the migration speed of the droplet in both the axial and cross-stream directions. This is in sharp contrast to a prediction of decelerated motion of the droplet under the same imposed thermal field, as obtained from asymptotic theory. We attribute this discrepancy to an alteration of the surface tension mediated interfacial stress due to the locally evolving temperature field, and a consequent concomitant alteration in the interfacial viscous stress to realize a tangential force balance at the interface. From scaling arguments, we show that the resulting change in the viscous drag force may occur over an order of magnitude, disrupting the outcome as compared to that obtained from asymptotic analysis. These results are likely to bear significant implications in controllable separation and sorting of deformable entities in confined fluidic media.

We simulated laminar flow through a hexagonal sphere pack driven by a sinusoidal volume force using direct numerical simulation. We vary two independent parameters, the Hagen and Womersley numbers, representing the amplitude and frequency of the forcing. First, we determine for which regions in the parameter space nonlinear effects have to be considered. We judge the presence of nonlinear effects from the departure of the superficial velocity and kinetic energy from a linear behaviour as well as from the presence of higher harmonics in the discrete Fourier transform of the velocity field. We discuss the asymptotic behaviour of the onset of nonlinearity in the limits of low and high Womersley number, and we delineate approximately the parameter region that can be described using the linear theory. Second, we document the changes of instantaneous velocity fields with Hagen and Womersley numbers. We show that the onset of nonlinearity is accompanied by a loss of fore–aft symmetry of the flow, and subsequently, we employ the deviation from this symmetry to quantify the strength of nonlinear effects in the instantaneous velocity fields. Based on this analysis, we demonstrate that for higher Womersley numbers, the strongest nonlinear effects occur during the deceleration of the superficial velocity; consequently, the development of the nonlinearity is not in phase with the superficial velocity. Finally, we describe the leading-order nonlinear effects in the frequency domain and the interaction among the nonlinear Fourier modes that leads to a temporal variation in the strength of nonlinear effects.

Modal global linear stability analysis of thermal convection is performed with the linearized lattice Boltzmann method (LLBM). The onset of Rayleigh–Bénard convection in rectangular cavities with conducting and adiabatic sidewalls and the instability of two-dimensional (2-D) and three-dimensional (3-D) natural convection in cavities are studied. The method of linearizing the local equilibrium probability distribution function that was first proposed by Pérez et al. (Theor. Comp. Fluid Dyn., vol. 31, 2017, pp. 643–664) is extended to solve the coupled linear Navier–Stokes equations together with the linear energy equation in this work. A multiscale analysis is also performed to recover the macroscopic linear Navier–Stokes equations from the discrete lattice Boltzmann equations for both the single and multiple relaxation time models. The present LLBM is implemented in the framework of the Palabos library. It is validated by calculating the linear critical value of 2-D natural convection that the LLBM with the multiple relaxation time model has an error less than 1 % compared with the spectral method. The instability mechanism of the flow is explained by kinetic energy transfer analysis. It is shown that the buoyancy mechanism and inertial mechanism tend to stabilize the Hopf bifurcation of the 2-D natural convection at Pr < 0.08 and Pr > 1, respectively. For 3-D natural convection, subcritical bifurcation of the Hopf type is found for low-Prandtl-number fluids (Pr < 0.1).

Dynamical constraints on the wall layer in turbulent pipe flow imply both a narrow peak in the streamwise component of the turbulent Lamb vector near the wall, and a scaling of the wall layer depth proportional to the depth of the viscous sublayer. An approximation of the Lamb vector distribution, which equates to the gradient of Reynolds stress, is proposed. Hence the equation for streamwise mean flow may be integrated to obtain an expression for the velocity profile in the wall layer.

Fully rough wall-sheared turbulence is composed of an inner and outer layer, the former occupied by sinuous structures sustained by the well-known autonomous inner cycle, the latter occupied by inclined parcels of momentum deviations relative to the average. The outer-layer structures are also known as uniform momentum zones (UMZs), where the presence of successive UMZs manifests instantaneously with a distinct ‘staircase’ pattern in streamwise velocity. Direct numerical simulation (DNS) of fully rough channel turbulence was recently used to interpret UMZs as wakes originating from antecedent bluff-body-like interactions within the inner layer (Anderson & Salesky, J. Fluid Mech., vol. 906, 2020, A8). Wake-scaling arguments agreed precisely with instantaneous results from DNS. Foremost among these results was evidence that the instantaneous wall-normal gradient of streamwise velocity exhibited scaling of ${\sim }x_3^{-1/2}$, where $x_3$ is wall-normal location; this is in contrast to logarithmic scaling, which requires the wall-normal gradient of streamwise velocity to scale as ${\sim }x_3^{-1}$. Herein, wake-scaling arguments have been advanced and compared against results from wall-modelled large-eddy simulation (LES). New results from the ‘bottom up’ wake-scaling arguments agree with LES results, such that the shear associated with the instantaneous staircase-like streamwise velocity follows a clear trend. We have leveraged conditional sampling during LES – predicated upon instantaneous low-frequency, high-magnitude surface stress values known to correspond to large-scale inertial-layer coherence – to further assess the predictive value of the wake-scaling arguments. Model results compare favourably.

Continuing from Part 1 (Hernández et al., J. Fluid Mech., vol. 936, 2022, A33), a generalised quasilinear (GQL) approximation is studied in turbulent channel flow using a flow decomposition defined with spanwise Fourier modes: the flow is decomposed into a set of low-wavenumber spanwise Fourier modes and the rest high-wavenumber modes. This decomposition leads to the nonlinear low-wavenumber group that supports the self-sustaining process within the given integral length scales, whereas the linearised high-wavenumber group is not able to do so, unlike the GQL models in Part 1, which place a minimal mathematical description for the self-sustaining process across all integral scales. Despite the important physical difference, it is shown that the GQL models in this study share some similarities with those in Part 1, i.e. the reduced multi-scale behaviour and anisotropic turbulent fluctuations. Furthermore, despite not being able to support the self-sustaining process in the high-wavenumber group, the GQL models in the present study are found to reproduce some key statistical features in the high-wavenumber group solely through the ‘scattering’ mechanism proposed by previous studies. Finally, using the nature of the GQL approximation, a further set of numerical experiments suppressing certain triadic nonlinear interactions are carried out. This unveils some key roles played by certain types of triadic interactions, including energy cascade and inverse energy transfer in the near-wall region. In particular, the inhibition of inverse energy transfer in the spanwise direction leads to suppression of the near-wall positive turbulent transport at large scales.

Development of the convective flow from a line heat source located at the liquid surface is studied for the case when thermocapillary convection is inhibited by the surface film. Surface film is formed by small amounts of surfactants present in non-deionized (distilled or natural) water. It imposes a no-slip boundary condition for velocity at the surface and determines the fluid dynamics of the surface layer. For the heat source located at the surface, the vertical stratification is stable and the flow is driven by a horizontal pressure gradient, formed by non-uniform heating of the liquid along the surface. Experimental studies include measurements of velocity fields using particle image velocimetry, surface relief measurements using moon-glade background oriented Schlieren and surface temperature measurements using infrared thermography. Also, numerical simulations of the flow development are performed and a new similarity solution for the steady state in a laterally unbounded domain is constructed, which takes into account the pressure gradient. The results are shown to be in good agreement with experimental data. The obtained dependencies of the maximal velocity, boundary layer thickness, local and total Nusselt numbers on the Rayleigh number are consistent with literature data obtained for the non-uniformly heated bottom case, which demonstrates universal properties of horizontal convection.

Large-scale magnetic fields thread through the electrically conducting matter of the interplanetary and interstellar medium, stellar interiors and other astrophysical plasmas, producing anisotropic flows with regions of high-Reynolds-number turbulence. It is common to encounter turbulent flows structured by a magnetic field with a strength approximately equal to the root-mean-square magnetic fluctuations. In this work, direct numerical simulations of anisotropic magnetohydrodynamic (MHD) turbulence influenced by such a magnetic field are conducted for a series of cases that have identical resolution, and increasing grid sizes up to $2048^3$. The result is a series of closely comparable simulations at Reynolds numbers ranging from 1400 up to 21 000. We investigate the influence of the Reynolds number from the Lagrangian viewpoint by tracking fluid particles and calculating single-particle and two-particle statistics. The influence of Alfvénic fluctuations and the fundamental anisotropy on the MHD turbulence in these statistics is discussed. Single-particle diffusion curves exhibit mildly superdiffusive behaviours that differ in the direction aligned with the magnetic field and the direction perpendicular to it. Competing alignment processes affect the dispersion of particle pairs, in particular at the beginning of the inertial subrange of time scales. Scalings for relative dispersion, which become clearer in the inertial subrange for a larger Reynolds number, can be observed that are steeper than indicated by the Richardson prediction.