We propose a model for the acceleration of micro-bubbles (smaller than the dissipative scale of the flow) subjected to the drag and fluid inertia forces in a homogeneous and isotropic turbulent flow. This model, that depends on the Stokes number, Reynolds number and the density ratio, reproduces the evolution of the acceleration variance as well as the relative importance and alignment of the two forces as observed from direct numerical simulations (DNS). We also report that the bubble acceleration statistics conditioned on the local kinetic energy dissipation rate are invariant with the Stokes number and the dissipation rate. Based on this observation, we propose a stochastic model for the instantaneous bubble acceleration vector accounting for the small-scale intermittency of the turbulent flows. The norm of the bubble acceleration is obtained by modelling the dissipation rate along the bubble trajectory from a log-normal stochastic process, whereas its orientation is given by two coupled random walks on a unit sphere in order to model the evolution of the joint orientation of the drag and inertia forces acting on the bubble. Furthermore, the proposed stochastic model for the bubble acceleration is used in the context of large eddy simulations (LES) of turbulent flows laden with small bubbles. To account for the turbulent motion at scales smaller than the mesh resolution, we decompose the instantaneous bubble acceleration in its resolved and residual parts. The first part is given by the drag and fluid inertia forces computed from the resolved velocity field, and the second term refers to the random contribution of small unresolved turbulent scales and is estimated with the stochastic model proposed in the paper. Comparisons with DNS and standard LES, show that the proposed model improves significantly the statistics of the bubbly phase.

Annular Couette flow is the flow between two coaxial cylinders driven by the axial translation of the inner cylinder. It is investigated using direct numerical simulation in long domains, with an emphasis on the laminar–turbulent coexistence regime found for marginally low values of the Reynolds number. Three distinct flow regimes are demonstrated as the radius ratio $\unicode[STIX]{x1D702}$ is decreased from 0.8 to 0.5 and finally to 0.1. The high-$\unicode[STIX]{x1D702}$ regime features helically shaped turbulent patches coexisting with laminar flow, as in planar shear flows. The moderate-$\unicode[STIX]{x1D702}$ regime does not feature any marked laminar–turbulent coexistence. In an effort to discard confinement effects, proper patterning is, however, recovered by artificially extending the azimuthal span beyond $2\unicode[STIX]{x03C0}$. Eventually, the low-$\unicode[STIX]{x1D702}$ regime features localised turbulent structures different from the puffs commonly encountered in transitional pipe flow. In this new coexistence regime, turbulent fluctuations are surprisingly short-ranged. Implications are discussed in terms of phase transition and critical scaling.

The propagation of a train of mode-2 internal solitary-like waves (ISWs) over a uniformly sloping, solid topographic boundary, has been studied by means of a combined laboratory and numerical investigation. The waves are generated by a lock-release method. Features of their shoaling include (i) formation of an oscillatory tail, (ii) degeneration of the wave form, (iii) wave run up, (iv) boundary layer separation, (v) vortex formation and re-suspension at the bed and (vi) a reflected wave signal. Slope steepness, $s$, is defined to be the height of the slope divided by the slope base length. In shallow slope cases ($s\leqslant 0.07$), the wave form is destroyed by the shoaling process; the leading mode-2 ISW degenerates into a train of mode-1 waves of elevation and little boundary layer activity is seen. For steeper slopes ($s\geqslant 0.13$), boundary layer separation, vortex formation and re-suspension at the bed are observed. The boundary layer dynamics is shown (numerically) to be dependent on the Reynolds number of the flow. A reflected mode-2 wave signal and wave run up are seen for slopes of steepness $s\geqslant 0.20$. The wave run up distance is shown to be proportional to the length scale $ac^{2}/g^{\prime }h_{2}\sin \unicode[STIX]{x1D703}$ where $a,c,g^{\prime },h_{2}$ and $\unicode[STIX]{x1D703}$ are wave amplitude, wave speed, reduced gravity, pycnocline thickness and slope angle respectively.

A comprehensive study of the fundamental characteristics of leading-edge separation in rarefied hypersonic flows is undertaken and its salient features are elucidated. Separation of a boundary layer undergoing strong expansion is typical in many practical hypersonic applications such as base flows of re-entry vehicles and flows over deflected control surfaces. Boundary layer growth under such conditions is influenced by effects of rarefaction and thermal non-equilibrium, thereby differing significantly from the conventional no-slip Blasius type. A leading-edge separation configuration presents a fundamental case for studying the characteristics of such a flow separation but with minimal influence from a pre-existing boundary layer. In this work, direct simulation Monte Carlo computations have been performed to investigate flow separation and reattachment in a low-density hypersonic flow over such a configuration. Distinct features of leading-edge flow, limited boundary layer growth, separation, shear layer, flow structure in the recirculation region and reattachment are all explained in detail. The fully numerical shear layer profile after separation is compared against a semi-theoretical profile, which is obtained using the numerical separation profile as the initial condition on existing theoretical concepts of shear layer analysis based on continuum flow separation. Experimental studies have been carried out to determine the surface heat flux using thin-film gauges and computations showed good agreement with the experimental data. Flow visualisation experiments using the non-intrusive planar laser-induced fluorescence technique have been performed to image the fluorescence of nitric oxide, from which velocity and rotational temperature distributions of the separated flow region are determined.

Fluid–structure interaction is fundamental to the characteristics of the induced flows due to the motion of structures in fluids and also is crucial to the performance of submerged structures. This paper presents a three-dimensional analytical study of the intrinsic free vibration of an elastic multilayered hollow sphere interacting with an exterior non-Newtonian fluid medium. The fluid is assumed to be characterized by a compressible linear viscoelastic model accounting for both the shear and compressional relaxation processes. For small-amplitude vibrations, the equations governing the viscoelastic fluid can be linearized, which are then solved by introducing appropriate potential functions. The solid is assumed to exhibit a particular material anisotropy, i.e. spherical isotropy, which includes material isotropy as a special case. The equations governing the anisotropic solid are solved in spherical coordinates using the state-space formalism, which finally establishes two separate transfer relations correlating the state vectors at the innermost surface with those at the outermost surface of the multilayered hollow sphere. By imposing the continuity conditions at the fluid–solid interface, two separate analytical characteristic equations are derived, which characterize two independent classes of vibration. Numerical examples are finally conducted to validate the theoretical derivation as well as to investigate the effects of various factors, including fluid viscosity and compressibility, fluid viscoelasticity, solid anisotropy and surface effect, as well as solid intrinsic damping, on the vibration characteristics of the submerged hollow sphere. Particularly, our theoretically predicted vibration frequencies and quality factors of gold nanospheres with intrinsic damping immersed in water agree exceptionally well with the available experimentally measured results. The reported analytical solution is truly and fully three-dimensional, covering from the purely radial breathing mode to the torsional mode to any general spheroidal mode as well as being applicable to various simpler situations, and hence can be a broad-spectrum benchmark in the study of fluid–structure interaction.

Langmuir circulation, a key turbulent process in the upper ocean, is mechanistically driven and sustained by imposed atmospheric wind stress and surface wave drift. In addition, and specifically in coastal zones, the presence of a mean current – whether associated with tidal currents or large-scale eddies – generates bottom-boundary-layer shear, which further modulates the physical attributes of coastal-zone Langmuir turbulence. We show that the presence of bottom-boundary-layer shear generated by oblique forcing between the mean current, atmospheric drag, and monochromatic wave field direction changes the orientation of the resultant, large-scale Langmuir cells. A model to predict this resultant orientation, based on salient parameters defining the forcing obliquity, is proposed. We also perform a systematic parametric study to isolate the ‘turning’ influence of salient parameters, which reveals that the resultant Langmuir cell orientation is always intermediate to the imposed forces. In order to provide a rigorous basis for the results, we study terms responsible for sustenance of streamwise vorticity, and provide a theoretical justification for the observed results.

The fundamental solution to steady ship waves accounting for viscous effects (the viscous-ship-wave Green function) is investigated within the framework of the weakly damped free-surface flow theory. An explicit expression of the viscous-ship-wave Green function is firstly derived, and an accurate and efficient technique is described to evaluate the Green function via decomposing the free-surface term into the local-flow component and wave component. To delve into the physical features of the viscous-ship-wave Green function, the asymptotic approximations in the far field due to Kelvin, Havelock and Peters are presented for the flow-field point located inside, at and outside the Kelvin wedge. In addition, uniform approximations to the wave component based on the Chester–Friedman–Ursell (CFU) approximation and the Kelvin–Havelock–Peters (KHP) approximation are carried out. Both numerical evaluation and asymptotic approximations show that the singular behaviour is eliminated and the divergent waves associated with large wavenumbers leading to rapid oscillations are severely damped when viscous effects are accounted for. In addition, viscous effects also alter the apparent wake angle associated with the wave pattern created by a high-speed translating source, and the apparent wake angle is dependent on both $\mathscr{U}^{-1}$ and $\mathscr{U}^{-2}$, where $\mathscr{U}$ is the translating speed of the source.

The influence of turbulence characteristics of a cross-airflow including its velocity fluctuations and integral length and time scales on the primary breakup regime, trajectory and breakup height and time of a transversely injected liquid jet was investigated experimentally. Turbulence intensity of the incoming airflow was varied from $u_{rms}/u_{g}=1.5\,\%$ to 5.5 % (where $u_{g}$ is cross-airflow streamwise mean velocity and $u_{rms}$ is the r.m.s. of the corresponding cross-airflow streamwise mean velocity fluctuation) by placing at the inlet of the test section a perforated plate/grid with a solidity ratio of $S=50\,\%$. Over the range of gas Weber number, $3.1<We_{g}<7.14$, the ensuing liquid jet exhibited more fluctuations and late breakup transitional behaviour under turbulent airflow conditions than in a uniform cross-airflow. Proper orthogonal decomposition of the liquid jet dynamics revealed that the use of grid caused a rise in the wavelength of travelling waves along the liquid jet, which hindered the transition of the liquid jet primary breakup regime from enhanced capillary breakup to the bag breakup mode. The quantitative results demonstrated that, at a constant airflow mean velocity, turbulent cross-airflow caused the liquid jet to bend earlier compared with its uniform counterpart. A power-law empirical correlation was proposed for the prediction of the liquid jet trajectory which takes into account the effect of turbulent Reynolds number. The liquid jet breakup height (in the $y$-axis direction) normalized by the jet diameter, and accordingly the liquid jet breakup time normalized by the airflow integral time scale, were found to decrease with increasing the airflow turbulence intensity. Two power-law empirical correlations were proposed to predict the liquid jet breakup height and time.

We study dynamo action in rotating, plane layer Boussinesq convection in the absence of inertia. This allows a decomposition of the velocity into a thermal part driven by buoyancy, and a magnetic part driven by the Lorentz force. We have identified three families of solutions, defined in terms of what is the dominant contribution to the velocity. In weak field dynamos the dominant contribution is the thermal component, in super strong field dynamos the dominant contribution is magnetic and in strong field dynamos the two components are comparable. For each of these solutions we investigate the force balance in the momentum equation to determine the relative importance of the viscous, buoyancy, Coriolis and magnetic forces. We do this by extracting the solenoidal part of the individual terms in the momentum equation, thereby removing their pressure contributions. This is numerically preferable to the more common practice of taking the curl of the momentum equation, which introduces an extra derivative. We find that, irrespective of the type of dynamo solution, the dynamics is controlled by the horizontal forces (in projection). Furthermore, in the progression from weak to strong to super strong dynamos, we find that the viscous forces in the thermal equation become negligible, thereby leading to a balance between buoyancy and Coriolis forces. On the other hand, no corresponding trend is observed in the magnetic part of the momentum equation: the viscous stresses always remain significant. This can be attributed to the different degrees of smoothness of the Coriolis and Lorentz forces, the latter having contributions from strong, filamentary structures. We discuss how our findings relate to dynamo solutions in which viscosity plays no role whatsoever – so-called Taylor states.

We present a generalisation of efficient numerical frameworks for modelling fluid–filament interactions via the discretisation of a recently developed, non-local integral equation formulation to incorporate regularised Stokeslets with half-space boundary conditions, as motivated by the importance of confining geometries in many applications. We proceed to utilise this framework to examine the drag on slender inextensible filaments moving near a boundary, firstly with a relatively simple example, evaluating the accuracy of resistive force theories near boundaries using regularised Stokeslet segments. This highlights that resistive force theories do not accurately quantify filament dynamics in a range of circumstances, even with analytical corrections for the boundary. However, there is the notable and important exception of movement in a plane parallel to the boundary, where accuracy is maintained. In particular, this justifies the judicious use of resistive force theories in examining the mechanics of filaments and monoflagellate microswimmers with planar flagellar patterns moving parallel to boundaries. We proceed to apply the numerical framework developed here to consider how filament elastohydrodynamics can impact drag near a boundary, analysing in detail the complex responses of a passive cantilevered filament to an oscillatory flow. In particular, we document the emergence of an asymmetric periodic beating in passive filaments in particular parameter regimes, which are remarkably similar to the power and reverse strokes exhibited by motile $9+2$ cilia. Furthermore, these changes in the morphology of the filament beating, arising from the fluid–structure interactions, also induce a significant increase in the hydrodynamic drag of the filament.