In this paper, we investigate the dynamics of an encapsulated bubble in steady and oscillatory electric fields theoretically, based on a leaky dielectric model. On the bubble surface, an applied electric field generates a Maxwell stress, in addition to hydrodynamic traction and membrane mechanical stress. Our model also includes the effect of interfacial charge due to the jump of the current and the stretching of the interface. We focus on the axisymmetric deformation of the encapsulated bubble induced by the electric field and carry out our analysis using Legendre polynomials. In our first example, the encapsulating membrane is modelled as a nearly incompressible interface with bending rigidity. Under a steady uniform electric field, the encapsulated bubble resumes an elongated equilibrium shape, dominated by the second- and fourth-order shape modes. The deformed shape agrees well with experimental observations reported in the literature. Our model reveals that the interfacial charge distribution is determined by the magnitude of the shape modes, as well as the permittivity and conductivity of the external and internal fluids. The effects of the electric field on the natural frequency of the oscillating bubble are also shown. For our second example, we considered a bubble encapsulated with a hyperelastic membrane with bending rigidity, subject to an oscillatory electric field. We show that the bubble can modulate its oscillating frequency and reach a stable shape oscillation at an appreciable amplitude.

We investigate the asymptotic properties of axisymmetric inertial modes propagating in a spherical shell when viscosity tends to zero. We identify three kinds of eigenmodes whose eigenvalues follow very different laws as the Ekman number $E$ becomes very small. First are modes associated with attractors of characteristics that are made of thin shear layers closely following the periodic orbit traced by the characteristic attractor. Second are modes made of shear layers that connect the critical latitude singularities of the two hemispheres of the inner boundary of the spherical shell. Third are quasi-regular modes associated with the frequency of neutral periodic orbits of characteristics. We thoroughly analyse a subset of attractor modes for which numerical solutions point to an asymptotic law governing the eigenvalues. We show that three length scales proportional to $E^{1/6}$, $E^{1/4}$ and $E^{1/3}$ control the shape of the shear layers that are associated with these modes. These scales point out the key role of the small parameter $E^{1/12}$ in these oscillatory flows. With a simplified model of the viscous Poincaré equation, we can give an approximate analytical formula that reproduces the velocity field in such shear layers. Finally, we also present an analysis of the quasi-regular modes whose frequencies are close to $\sin (\unicode[STIX]{x03C0}/4)$ and explain why a fluid inside a spherical shell cannot respond to any periodic forcing at this frequency when viscosity vanishes.

The present study is devoted to characterizing the coherent organization of vortical structures, which can be fitted into the paradigm of the hairpin-packet model, in the streamwise–wall-normal plane of a canonical turbulent boundary layer at $Re_{\unicode[STIX]{x1D70F}}=377{-}1093$. Proper orthogonal decomposition (POD) of the planar velocity fields measured via two-dimensional particle image velocimetry, together with a spatio-temporal coherence analysis, shows that the first four leading-order POD modes share both geometric similarity and dynamic coherence and jointly depict the downstream convection of the large-scale Q2/Q4 events, which can be regarded as the low-order imprints of the hairpin packets. A simple low-order indicator is then proposed to extract the inclined interfaces of the hairpin packets, based on which a two-point conditional correlation analysis forms a statistical picture of the spatial organization of multiple prograde vortices aligned along the interface within one packet. A saturation of the self-similar growth of the streamwise gap between two neighbouring vortices is seen. This implies a detachment of the hairpin packets from the inner layer. Both the detachment height and the saturated streamwise spacing are found to scale as $Re_{\unicode[STIX]{x1D70F}}^{1/2}$.

The laminar–turbulent transition of a plane channel entrance flow is revisited using global linear optimization analyses and direct numerical simulations. The investigated case corresponds to uniform upstream velocity conditions and a moderate value of Reynolds number so that the two-dimensional developing flow is linearly stable under the parallel flow assumption. However, the boundary layers in the entry zone are capable of supporting the development of streaks, which may experience secondary instability and evolve to turbulence. In this study, global optimal linear perturbations are computed and studied in the nonlinear regime for different values of streak amplitude and optimization time. These optimal perturbations take the form of wavepackets having either varicose or sinuous symmetry. It is shown that, for short optimization times, varicose wavepackets grow through a combination of Orr and lift-up effects, whereas for longer target times, both sinuous and varicose wavepackets exhibit an instability mechanism driven by the presence of inflection points in the streaky flow. In addition, while the optimal varicose modes obtained for short optimization times are localized near the inlet, where the base flow is strongly three-dimensional, when the target time is increased, the sinuous and varicose optimal modes are displaced farther downstream, in the nearly parallel streaky flow. Finally, the optimal wavepackets are found to lead to turbulence for sufficiently high initial amplitudes. It is noticed that the resulting turbulent flows have the same wall-shear stress, whether the wavepackets have been obtained for short or for long time optimization.

Laboratory experiments were performed on a geometrically scaled vertical-axis wind turbine model over an unprecedented range of Reynolds numbers, including and exceeding those of the full-scale turbine. The study was performed in the high-pressure environment of the Princeton High Reynolds number Test Facility (HRTF). Utilizing highly compressed air as the working fluid enabled extremely high Reynolds numbers while still maintaining dynamic similarity by matching the tip speed ratio (defined as the ratio of tip velocity to free stream, $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D714}R/U$) and Mach number (defined at the turbine tip, $Ma=\unicode[STIX]{x1D714}R/a$). Preliminary comparisons are made with measurements from the full-scale field turbine. Peak power for both the field data and experiments resides around $\unicode[STIX]{x1D706}=1$. In addition, a systematic investigation of trends with Reynolds number was performed in the laboratory, which revealed details about the asymptotic behaviour. It was shown that the parameter that characterizes invariance in the power coefficient was the Reynolds number based on blade chord conditions ($Re_{c}$). The power coefficient reaches its asymptotic value when $Re_{c}>1.5\times 10^{6}$, which is higher than what the field turbine experiences. The asymptotic power curve is found, which is invariant to further increases in Reynolds number.

The stability of the wakes of cylinders with triangular cross-sections at incidence is investigated using Floquet stability analysis to elucidate the effects of cylinder inclination on the dominant flow instability. The upper limit of the Reynolds numbers (scaled by the height projected by the cylinder in this study) at which the wake of the two-dimensional base flow is time periodic is $Re\approx 140$ for most cylinder inclinations, exceeding which the flow becomes aperiodic, restricting the range of Reynolds numbers permitted for the stability analysis. Two different instability modes are predicted to manifest as the first-occurring mode at various cylinder inclinations – a regular mode possessing perturbation structures consistent with mode A dominates the wakes of cylinders at inclinations $\unicode[STIX]{x1D6FC}\lesssim 34.6^{\circ }$ and $\unicode[STIX]{x1D6FC}\gtrsim 55.4^{\circ }$, with a subharmonic mode consistent with mode C emerging as the primary mode in the wakes of the cylinder at the intermediate range of inclinations. For all inclinations, the mode B branch is not detected within the range of Reynolds numbers examined. The peak instability growth rates corresponding to mode A for all cylinder inclinations describe a linear variation with $(Re-Re_{A})/Re_{A}$, where $Re_{A}$ is the mode A transition Reynolds number, while those corresponding to mode C vary only approximately linearly. The generalized trend most pertinently shows mode C to develop more rapidly than mode A at inclinations which permit it. Examination of the near wake of the two-dimensional time-periodic base flow demonstrates the dependence of the development and intensity of mode C on imbalances in the flow solution over each shedding period, directly implying that the two-dimensional base flow solutions deviate from the half-period-flip map as the cylinder inclination is increased. The degree of asymmetry of the two-dimensional base flow relative to the ideal half-period-flip map is quantified using several measures. The results show distinctly different trends in these asymmetry measures between inclinations where modes A or C are dominant, agreeing with results from the stability analysis. The nature of the predicted instability modes at transition are also investigated by applying the Stuart–Landau equation, showing the transitions to be supercritical for all cylinder inclinations, with mode C being consistently more strongly supercritical than mode A.

The problem of transonic flow past an array of micro-electro-mechanical-type (MEMS-type) heating elements placed on a flat surface is investigated using the triple-deck theory. The compressible Navier–Stokes equations supplemented by the energy equation are considered for large Reynolds numbers. The triple-deck problem is formulated with the aid of the method of matched expansions. The resulting nonlinear viscous lower deck problem, coupled with the upper deck problem governed by the nonlinear Kármán–Guderley equation, is solved using a numerical method based on Chebyshev collocation and finite differences. Our results show the differences in subsonic and supersonic flow behaviour over heated elements. The results indicate the possibility of using the elements to favourably control the transonic flow field.

Kinetic equations are widely used in many branches of science to describe the evolution of random wave spectra. To examine the validity of these equations, we study numerically the long-term evolution of water wave spectra without wind input using three different models. The first model is the classical kinetic (Hasselmann) equation (KE). The second model is the generalised kinetic equation (gKE), derived employing the same statistical closure as the KE but without the assumption of quasistationarity. The third model, which we refer to as the DNS-ZE, is a direct numerical simulation algorithm based on the Zakharov integrodifferential equation, which plays the role of the primitive equation for a weakly nonlinear wave field. It does not employ any statistical assumptions. We perform a comparison of the spectral evolution of the same initial distributions without forcing, with/without a statistical closure and with/without the quasistationarity assumption. For the initial conditions, we choose two narrow-banded spectra with the same frequency distribution and different degrees of directionality. The short-term evolution ($O(10^{2})$ wave periods) of both spectra has been previously thoroughly studied experimentally and numerically using a variety of approaches. Our DNS-ZE results are validated both with existing short-term DNS by other methods and with available laboratory observations of higher-order moment (kurtosis) evolution. All three models demonstrate very close evolution of integral characteristics of the spectra, approaching with time the theoretical asymptotes of the self-similar stage of evolution. Both kinetic equations give almost identical spectral evolution, unless the spectrum is initially too narrow in angle. However, there are major differences between the DNS-ZE and gKE/KE predictions. First, the rate of angular broadening of initially narrow angular distributions is much larger for the gKE and KE than for the DNS-ZE, although the angular width does appear to tend to the same universal value at large times. Second, the shapes of the frequency spectra differ substantially (even when the nonlinearity is decreased), the DNS-ZE spectra being wider than the KE/gKE ones and having much lower spectral peaks. Third, the maximal rates of change of the spectra obtained with the DNS-ZE scale as the fourth power of nonlinearity, which corresponds to the dynamical time scale of evolution, rather than the sixth power of nonlinearity typical of the kinetic time scale exhibited by the KE. The gKE predictions fall in between. While the long-term DNS show excellent agreement with the KE predictions for integral characteristics of evolving wave spectra, the striking systematic discrepancies for a number of specific spectral characteristics call for revision of the fundamentals of the wave kinetic description.

The orientation of spheroidal particles dispersed in a fluid flow is known to influence the particles’ rotation mode. Rod-like and disk-like particles orient themselves differently and accordingly also rotate differently. In order to explore the role of the deterministic factors, i.e. mean shear and vorticity anisotropy, on the orientational behaviour of inertialess tracer spheroids, we adopted a purpose-made Couette–Poiseuille flow simulated numerically by Yang et al. (Intl J. Heat Fluid Flow, vol. 63, 2017, pp. 14–27). Typical wall turbulence with streamwise-oriented streaky structures caused by the locally high mean shear rate was observed only at one of the walls. The absence of mean shear at the other wall gave rise to an atypical turbulence field. Over a relatively wide and quasi-homogeneous core region, a modest mean shear rate made the vorticity field anisotropic. In spite of the mean shear, rod-like tracers were spinning and disk-like spheroids were tumbling, just as in homogeneous isotropic turbulence. We explained this phenomenon by the isotropic particle orientation and concluded that zero mean shear is not necessary for rod spinning and disk tumbling. The orientation and rotation of the Lagrangian tracer spheroids near the high shear wall were almost indistinguishable from the well-known behaviour in turbulent channel flows. Near the opposite wall, where the mean shear was negligibly small, disk-like particles aligned preferentially in the wall-normal direction and rotated similarly as in the presence of high shear. Rod-like particles, on the contrary, aligned more randomly and accordingly rotated similarly as in the core region. These observations revealed that the degree of particle orientation anisotropy has a major impact on the particle rotation mode, whereas mean shear, irrespective of the actual shear rate, only plays a secondary role in particle rotation. Deduction of the eigenvectors of the left Cauchy–Green tensor showed that the preferential orientation of the tracer spheroids was caused by the alignment of rods and disks with the strongest Lagrangian stretching and compression directions, respectively. Lagrangian stretching/compression determines the particle orientations and the particle orientation affects the particle rotation mode.

The spreading of evaporating drops without a pinned contact line is studied experimentally and theoretically, measuring the radius $R(t)$ of completely wetting alkane drops of different volatility on glass. Initially the drop spreads ($R$ increases), then owing to evaporation reverses direction and recedes with an almost constant non-zero contact angle $\unicode[STIX]{x1D703}\propto \unicode[STIX]{x1D6FD}^{1/3}$, where $\unicode[STIX]{x1D6FD}$ measures the rate of evaporation; eventually the drop vanishes at a finite-time singularity. Our theory, based on a first-principles hydrodynamic description, well reproduces the dynamics of $R$ and the value of $\unicode[STIX]{x1D703}$ during retraction.

Naturally developing and acoustically excited shear layers at the Reynolds numbers $Re_{\unicode[STIX]{x1D703}_{0}}=U\unicode[STIX]{x1D703}_{0}/\unicode[STIX]{x1D708}=85{-}945$ are studied using the hot-wire (HW) anemometry and particle image velocimetry (PIV), with a focus on the shear/rotation competition during the initial Kelvin–Helmholtz (KH) roll-up. Velocity spectra and the spatial linear stability (LST) analysis characterize the fundamental ($f_{n}$) and its subharmonic ($f_{n}/2$) mode interacting due to the vortex pairing. For $276\leqslant Re_{\unicode[STIX]{x1D703}_{0}}\leqslant 780$, the root-mean-square (r.m.s.) of the streamwise turbulence intensity shows a double-peaking phenomenon, i.e. major and minor peaks of the $u_{rms}$ coexist towards the high-speed (HS) and the low-speed (LS) sides, respectively. The single/double-peaked $u_{rms}$ profiles are found to be correlated with the scattered/organized distribution of the shear/rotation, demonstrating a transitioning character with the downstream distance, $Re_{\unicode[STIX]{x1D703}_{0}}$ and the upstream turbulence levels. The rotating vortex cores and the corresponding peripheral shear regions, demonstrate the phase reversal of the velocity fluctuations with respect to the HS and the LS sides. Excitation at $f_{n}$ increases the vortex count by 21 %, advances the location of the first KH roll-up and hence also the minor peak formation location. Due to the enhanced pairing at the $f_{n}/2$ forcing, the vortex count reduces by 23 %. Before merging into the downstream rotation core, the upstream vortex is shifted towards the HS side and the major peak is accordingly augmented. Actuation advances the transition to the nonlinear state, as well as the saturation of the amplification factor. The volumetric topologies of the shear/rotation loops tracked in consecutive phases during the period of the acoustic excitation, separate from the edge and grow in time–space due to the viscous diffusion. The shearing and rotating loops are found to be associated with the thinning (elongation) and expansion (accumulation) of the vorticity, respectively.

The transition to unsteadiness of a three-dimensional open cavity flow is investigated using the joint application of direct numerical simulations and fully three-dimensional linear stability analyses, providing a clear understanding of the first two bifurcations occurring in the flow. The first bifurcation is characterized by the emergence of Taylor–Görtler-like vortices resulting from a centrifugal instability of the primary vortex core. Further increasing the Reynolds number eventually triggers self-sustained periodic oscillations of the flow in the vicinity of the spanwise end walls of the cavity. This secondary instability causes the emergence of a new set of Taylor–Görtler vortices experiencing a spanwise drift directed toward the spanwise end walls of the cavity. While a two-dimensional stability analysis would fail to capture this secondary instability due to the neglect of the lateral walls, it is the first time to our knowledge that this drifting of the vortices can be entirely characterized by a three-dimensional linear stability analysis of the flow. Good agreements with experimental observations and measurements strongly support our claim that the initial stages of the transition to turbulence of three-dimensional open cavity flows are solely governed by modal instabilities.

The effects of isolated, cylindrical roughness elements on laminar–turbulent transition in a flat-plate boundary layer are investigated in a laminar water channel. Our experiments aim at providing a comparison to global linear stability theory (LST) by means of hot-film anemometry and particle image velocimetry. Although the critical Reynolds number from theory does not match the transition Reynolds number observed in experiments, there are distinct experimental observations indicating a changeover from purely convective to absolute/global instability very close to the critical Reynolds number predicted by theory. Forcing with a vibrating wire reveals the evolution of the system dynamics from an amplifier to a wavemaker when the critical Reynolds number is exceeded. The mode symmetry is varicose for thick roughness elements and a changeover from varicose to sinuous modes is observed at the critical Reynolds number for thin roughness elements. Therefore, most predictions by global LST can be confirmed, but additional observations in the physical flow demonstrate that not all features can be captured adequately by global LST.

Slender liquid jets that have a curved trajectory have been examined in a range of papers using a method introduced in Wallwork et al. (Proc. IUTAM Symp. on Free-Surface Flows, 2000, Kluwer; J. Fluid Mech., vol. 459, 2002, pp. 43–65) and Decent et al. (J. Engng Maths, vol. 42, 2002, pp. 265–282), for jets that emerge from an orifice on the surface of a rotating cylindrical container, successfully comparing computational results to measurements arising from laboratory experiments. Wallwork et al. (2000, 2002) and Decent et al. (2002) based their analyses on the slenderness of the jet, and neglected the torsion of the centreline of the jet, which is valid since in most situations examined the torsion is zero or small. Shikhmurzaev & Sisoev (J. Fluid Mech., vol. 819, 2017, pp. 352–400) used differential geometry and incorporated the torsion. This paper shows that these two methods produce identical results at leading order when the torsion is zero or when the torsion is $O(1)$, in an asymptotic framework based upon the slenderness of the jet, and shows that the method of Wallwork et al. (2000, 2002) and Decent et al. (2002) is accurate for parameters corresponding to scenarios previously examined and also when the torsion is $O(1)$. It is shown that the method of Shikhmurzaev & Sisoev (2017) should be used when the torsion is asymptotically large or when the jet is not slender.

The mechanism underlying the coherent hairpin process in wall-bounded shear flows is studied. An algorithm for the identification and analysis of flow structures based on morphological operations is presented. The method distils the topology of the flow field into a discrete data set and enables the time-resolved sampling of coherent flow processes across multiple scales. Application to direct simulation data of transitional and turbulent boundary layers at moderate Reynolds number sheds light on the flow physics of the hairpin process. The analysis links the hairpin to an exponential instability which is amplified in the flow distorted by a negative perturbation in the streamwise velocity component. Linear analyses substantiate the connection to an inviscid instability mechanism of varicose type. The formation of packets of hairpins is related to a self-similar process which originates from a single patch of low-speed fluid and describes a recurrence of the dynamics that leads to the formation of an individual hairpin. Analysis of the evolution of several thousand turbulent hairpins provides a statistical characterization of the principal dynamics and yields a time-resolved average of the hairpin process. Comparisons with the transitional hairpin show qualitatively consistent trends and thus support earlier hypotheses of their equivalence. In terms of the causality of events, the results suggest that the hairpin is a manifestation of the varicose instability and as such is a consequence rather than a cause of the primary perturbations of the flow.

Equilibrium and non-equilibrium intrusions released from long, full-depth locks and advancing into linearly stratified ambients are investigated via a vorticity-based modelling approach, for the constant-velocity slumping phase. Non-equilibrium intrusions give rise to an upstream-propagating wave, which we model as a bore. Predictions by the vorticity model agree closely with two-dimensional direct numerical simulation results and earlier experimental and computational data regarding front velocities, current thicknesses as well as the height and propagation velocity of the bore.

It is well known that the computation of the Basset-like history force is very demanding in terms of CPU and memory requirements, since it requires the evaluation of a history integral. We use the recent rational theory of Beylkin & Monzón (Appl. Comput. Harmon. Anal., vol. 19, 2005, pp. 17–48) to approximate the history kernel in the form of exponential sums to reformulate the viscous history force in a differential form. This theory allows us to approximate the history kernel in terms of exponential sums to any desired order of accuracy. This removes the need for long-time storage of the acceleration histories of the particle and the fluid. The proposed differential form approximation is applied to compute the history force on a spherical particle in a synthetic turbulent flow and a wall-bounded turbulent channel flow. Particles of various diameters are considered, and results obtained using the present technique are in reasonable agreement with those achieved using the full history integral.

The vorticity-based modelling approach originally introduced by Borden & Meiburg (J. Fluid Mech., vol. 726, 2013b, R1) is extended to gravity currents propagating into two-layer stratified ambients. Vorticity models are developed for three different flow configurations: no upstream-propagating wave, an upstream-propagating expansion wave only and an upstream-propagating expansion wave and a bore. For a given gravity current height and stratification strength, along with ambient inflow layer thicknesses and velocities, the models yield the gravity current velocity, the bore and expansion wave properties and the ambient outflow layer thicknesses and velocities. We furthermore establish which of the three configurations will occur in a given parameter regime. Since energy-related closure assumptions are not required for any of the configurations, we can determine the dissipation as a function of the gravity current height, for a given set of flow parameters. To investigate which gravity current height is selected in real flows, we carry out two-dimensional Navier–Stokes simulations for comparison. These yield gravity current heights close to the vorticity model solutions for energy-conserving flows. Hence we adopt these energy-conserving solutions as the vorticity model predictions. We subsequently discuss these predictions in the context of earlier models by other authors, and of two-layer stratified flows over obstacles.

When a contact line moves with a sufficiently large speed, liquid or gas films can be entrained on a solid depending on the direction of contact-line movement. In this work, the contact-line dynamics in the situation of a generic two-fluid system is investigated. We demonstrate that the hydrodynamics of a contact line, no matter whether advancing or receding, can formally reduce to that of a receding one with small interfacial slopes. Since the latter can be well treated under the classical lubrication approximation, this analogy allows us to derive an asymptotic solution of the interfacial profiles for arbitrary values of contact angle and viscosity ratio. For the dip-coating geometry, we obtain, with no adjustable parameters, an analytical formula for the critical speed of wetting transition, which in particular predicts the onset of both liquid and gas entrainment. Moreover, the present analysis also builds a novel connection between the Cox–Voinov law and classical lubrication theory for moving contact lines.

In this paper, an analytical investigation of two-dimensional conventional Burnett equations has been undertaken for gaseous flow through a long microchannel. The analytical solution is obtained by using perturbation analysis around the classical Navier–Stokes solution with appropriate boundary conditions. The perturbation expansion is employed with the smallness parameter $\unicode[STIX]{x1D716}$, taken as the ratio of height to length of the microchannel. The solution for pressure is obtained by solving the cross-stream momentum equation while the velocity distribution is obtained from the streamwise momentum equation. The resulting ordinary differential equations in pressure and velocity are third-order and second-order, respectively. The required boundary conditions for pressure are obtained from direct simulation Monte Carlo (DSMC) data. The obtained analytical solution matches the available DSMC solution well. This is perhaps the first analytical solution of the Burnett equations using the perturbation approach.