The paper by Dritschel et al. (J. Fluid Mech., vol. 783, 2015, pp. 1–22) describes the long-time behaviour of inviscid two-dimensional fluid dynamics on the surface of a sphere. At issue is whether the flow settles down to an equilibrium or whether, for generic (random) initial conditions, the long-time solution is periodic, quasi-periodic or chaotic. While it might be surprising that this issue is not settled in the literature, it is important to keep in mind that the Euler equations form a dissipationless Hamiltonian system, hence the set of equations only redistributes the initial vorticity, generating smaller and smaller scales, while keeping kinetic energy, angular impulse and an infinite family of vorticity moments (Casimirs) intact. While special solutions that never settle down to an equilibrium state can be constructed using point vortices, vortex patches and other distributions, the fate of random initial conditions is a trickier problem. Previous statistical theories indicate that the long-time state should be a stationary large-scale distribution of vorticity. By carrying out careful numerical simulations using two different methods, the authors make a compelling case that the generic long-time state resembles a large-scale oscillating quadrupolar vorticity field, surrounded by persistent small-scale vortices. While numerical simulations can never conclusively settle this issue, the results might help guide future theories that seek to prove the existence of such an interesting dynamical long-time state.

Rotating-disk boundary-layer flow is known to be locally absolutely unstable at $R>507$ as shown by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) and, for the clean-disk condition, experimental observations show that the onset of transition is highly reproducible at that Reynolds number. However, experiments also show convectively unstable stationary vortices due to cross-flow instability triggered by unavoidable surface roughness of the disk. We show that if the surface is sufficiently rough, laminar–turbulent transition can occur via a convectively unstable route ahead of the onset of absolute instability. In the present work we compare the laminar–turbulent transition processes with and without artificial surface roughnesses. The differences are clearly captured in the spectra of velocity time series. With the artificial surface roughness elements, the stationary-disturbance component is dominant in the spectra, whereas both stationary and travelling components are represented in spectra for the clean-disk condition. The wall-normal profile of the disturbance velocity for the travelling mode observed for a clean disk is in excellent agreement with the critical absolute instability eigenfunction from local theory; the wall-normal stationary-disturbance profile, by contrast, is distinct and the experimentally measured profile matches the stationary convective instability eigenfunction. The results from the clean-disk condition are compared with theoretical studies of global behaviours in spatially developing flow and found to be in good qualitative agreement. The details of stationary disturbances are also discussed and it is shown that the radial growth rate is in excellent agreement with linear stability theory. Finally, large stationary structures in the breakdown region are described.

The stability of a thin flexible plate confined inside an inviscid two-dimensional channel is examined using a nonlinear eigenvalue analysis method. A new Green’s function for the vortex wake of the flexible plate inside the channel, as well as its rapidly convergent series approximation, is proposed. Comparison with a fully coupled Navier–Stokes fluid–structure interaction model indicates that the current inviscid model successfully predicts the flutter boundary for a confined flexible plate. The analysis also shows that confinement has a destabilizing effect on heavy plates. Furthermore, as the confinement is increased, the oscillating frequency of the plate increases and new peaks appear in its stability curve. Asymmetric placement of the plate within the channel, especially when the plate is very close to one wall, also modifies the stability curve of the system by shifting the mode transition points toward smaller fluid-to-plate inertia ratios. Our study suggests that the degree of confinement and asymmetric placement of the plate in the channel could be used to alter the flutter instability of the plate, and to adjust the frequency of flutter.

Interfacial fluid mixing driven by an external acceleration or a shock wave are common phenomena known as Rayleigh–Taylor instability and Richtmyer–Meshkov instability, respectively. The most significant feature of these instabilities is the penetrations of heavy (light) fluid into light (heavy) fluid known as spikes (bubbles). The study of the growth rate of these fingers is a classical problem in fundamental science and has important applications. Research on this topic has been very active over the past half-century. In contrast to the well-known phenomena that spikes and bubbles can have quantitatively, even qualitatively, different behaviours, we report a surprising result for fingers in a two-dimensional system: in terms of scaled dimensionless variables, all spikes and bubbles at any density ratio closely follow a universal curve, up through a pre-asymptotic stage. Such universality holds not only among bubbles and among spikes of different density ratios, but also between bubbles and spikes of different density ratios. The data from numerical simulations show good agreement with our theoretical predictions.

In this paper we present a theoretical model to study sound scattering from flow ducts with a semi-infinite lining surface covered by some equally spaced rigid splices, which is of practical importance in the development of silent aeroengines. The key contribution of our work is the analytical and rigorous description of axial liner splices by incorporating Fourier series expansion and the Wiener–Hopf method. In particular, we describe periodic variations of the semi-infinite lining surface by using Fourier series that accurately represent the layout of rigid splices in the circumferential direction. The associated matrix kernel involves a constant matrix and a diagonal matrix. The latter consists of a series of typical scalar kernels. A closed-form solution is then obtained by using standard routines of Wiener–Hopf factorisation for scalar kernels. A couple of appropriate approximations, such as numerical truncations of infinite Fourier series, have to be adopted in the implementation of this theoretical model, which is validated by comparing favorably with numerical solutions from a commercial acoustic solver. Finally, several numerical test cases are performed to demonstrate this theoretical model. It can be seen that the proposed theoretical model helps to illuminate the essential acoustic effect jointly imposed by axial and circumferential hard–soft interfaces.

We employ the moderate-Péclet-number macroscale model developed in part 2 of this sequence (Schnitzer et al., J. Fluid Mech., vol. 704, 2012, pp. 109–136) towards the calculation of electroviscous forces on charged solid particles engendered by an imposed relative motion between these particles and the electrolyte solution in which they are suspended. In particular, we are interested in the kinematic irreversibility of these forces, stemming from the diffusio-osmotic slip which accompanies the salt-concentration polarisation induced by that imposed motion. We illustrate the electroviscous irreversibility using two prototypic problems, one involving side-by-side sedimentation of two spherical particles, and the other involving a force-free spherical particle suspended in the vicinity of a planar wall and exposed to a simple shear flow. We focus on the pertinent limit of near-contact configurations, where use of lubrication approximations provides closed-form expressions for the leading-order lateral repulsion. In this approximation scheme, the need to solve the advection–diffusion equation governing the salt-concentration polarisation is circumvented.

We present a numerical analysis of the rheology of a dense suspension of spherical capsules in simple shear flow in the Stokes flow regime. The behaviour of neo-Hookean capsules is simulated for a volume fraction up to ${\it\phi}=0.4$ by graphics processing unit computing based on the boundary element method with a multipole expansion. To describe the specific viscosity using a polynomial equation of the volume fraction, the coefficients of the equation are calculated by least-squares fitting. The results suggest that the effect of higher-order terms is much smaller for capsule suspensions than rigid sphere suspensions; for example, $O({\it\phi}^{3})$ terms account for only 8 % of the specific viscosity even at ${\it\phi}=0.4$ for capillary numbers $Ca\geqslant 0.1$. We also investigate the relationship between the deformation and orientation of the capsules and the suspension rheology. When the volume fraction increases, the deformation of the capsules increases while the orientation angle of the capsules with respect to the flow direction decreases. Therefore, both the specific viscosity and the normal stress difference increase with volume fraction due to the increased deformation, whereas the decreased orientation angle suppresses the specific viscosity, but amplifies the normal stress difference.

The large-amplitude oscillatory flow of a suspension of spherical particles in a pipe is studied at low Reynolds number. Particle volume fraction and velocity are examined through refractive index matching techniques. The particles migrate toward the centre of the pipe, i.e. toward regions of lower shear rate, for bulk volume fractions larger than 10 %. Steady results are in agreement with available experimental results and discrete-particle simulations for similar geometries. The dynamics of the shear-induced migration process are analysed and compared against the predictions of the suspension balance model using realistic rheological laws.

In this paper we analyse the process of the generation of Tollmien–Schlichting waves in a laminar boundary layer on an aircraft wing in the transonic flow regime. We assume that the boundary layer is exposed to a weak acoustic noise. As it penetrates the boundary layer, the Stokes layer forms on the wing surface. We further assume that the boundary layer encounters a local roughness on the wing surface in the form of a gap, step or hump. The interaction of the unsteady perturbations in the Stokes layer with steady perturbations produced by the wall roughness is shown to lead to the formation of the Tollmien–Schlichting wave behind the roughness. The ability of the flow in the boundary layer to convert ‘external perturbations’ into instability modes is termed the receptivity of the boundary layer. In this paper we first develop the linear receptivity theory. Assuming the Reynolds number to be large, we use the transonic version of the viscous–inviscid interaction theory that is known to describe the stability of the boundary layer on the lower branch of the neutral curve. The linear receptivity theory holds when the acoustic noise level is weak, and the roughness height is small. In this case we were able to deduce an analytic formula for the amplitude of the generated Tollmien–Schlichting wave. In the second part of the paper we lift the restriction on the roughness height, which allows us to study the flows with local separation regions. A new ‘direct’ numerical method has been developed for this purpose. We performed the calculations for different values of the Kármán–Guderley parameter, and found that the flow separation leads to a significant enhancement of the receptivity process.

We determine approximate analytical solutions for average periodic trajectories of particles that are accelerated by the turbulent shearing of a fluid between collisions with a hydrodynamically rough bed. We indicate how the viscosity of the fluid may influence the collisions with the bed. The approximate solutions compare well with periodic solutions for average periodic trajectories over rigid bumpy and erodible beds that are generated numerically. The analytic solutions permit the determination of the relations between the particle flux and the strength of the shearing flow over a range of particle and fluid properties that vary between those for sand in air and sand in water.

We present direct numerical simulations (DNS) of unforced stratified turbulence with the objective of testing the strongly stratified turbulence theory. According to this theory the characteristic vertical scale of the turbulence is given by $\ell _{v}\sim u_{h}/N$, where $u_{h}$ is the horizontal velocity scale and $N$ the Brunt–Väisälä frequency. Combined with the hypothesis of the energy dissipation rate scaling as ${\it\epsilon}\sim u_{h}^{3}/\ell _{h}$, this theory predicts inertial range scalings for the horizontal spectrum of horizontal kinetic energy and of potential energy, according to $E(k_{h})\propto k_{h}^{-5/3}$. We begin by presenting a scaling analysis of the horizontal vorticity equation from which we recover the result regarding the vertical scale, $\ell _{v}\sim u_{h}/N$, highlighting in the process the important dynamical role of large-scale vertical shear of horizontal velocity. We then present the results from decaying DNS, which show a good agreement with aspects of the theory. In particular, the vertical Froude number is found to reach a constant plateau in time, of the form $Fr_{v}=u_{h}/(N\ell _{v})=C$ with $C=O(1)$ in all the runs. The derivation of the dissipation scaling ${\it\epsilon}\sim u_{h}^{3}/\ell _{h}$ at low Reynolds number in the context of decaying stratified turbulence highlights that the same scaling holds at high $\mathscr{R}=ReFr_{h}^{2}\gg 1$ as well as at low $\mathscr{R}\ll 1$, which is known (see Brethouwer et al., J. Fluid Mech., vol. 585, 2007, pp. 343–368) but not sufficiently emphasized in recent literature. We find evidence in our DNS of the dissipation scaling holding at $\mathscr{R}=O(1)$, which we interpret as being in the viscous regime. We also find ${\it\epsilon}_{k}\sim u_{h}^{3}/\ell _{h}$ and ${\it\epsilon}_{p}\sim u_{h}^{3}/\ell _{h}$ (with ${\it\epsilon}={\it\epsilon}_{k}+{\it\epsilon}_{p}$), in our high-resolution run at earlier times corresponding to $\mathscr{R}=O(10)$, which is in the transition between the strongly stratified and the viscous regimes. The horizontal spectrum of horizontal kinetic energy collapses in time using the scaling $E_{h}(k_{h})=C_{1}{\it\epsilon}_{k}^{2/3}k_{h}^{-5/3}$ and the horizontal potential energy spectrum is well described by $E_{p}(k_{h})=C_{2}{\it\epsilon}_{p}{\it\epsilon}_{k}^{-1/3}k_{h}^{-5/3}$. The presence of an inertial range in the horizontal direction is confirmed by the constancy of the energy flux spectrum over narrow ranges of $k_{h}$. However, the vertical energy spectrum is found to differ significantly from the expected $E_{h}(k_{v})\sim N^{2}k_{v}^{-3}$ scaling, showing that $Fr_{v}$ is not of order unity on a scale-by-scale basis, thus providing motivation for further investigation of the vertical structure of stratified turbulence.

Well-resolved measurements of the small-scale dissipation statistics within turbulent channel flow are reported for a range of Reynolds numbers from $Re_{{\it\tau}}\approx 500$ to 4000. In this flow, the local large-scale Reynolds number based on the longitudinal integral length scale is found to poorly describe the Reynolds number dependence of the small-scale statistics. When a length scale based on Townsend’s attached-eddy hypothesis is used to define the local large-scale Reynolds number, the Reynolds number scaling behaviour was found to be more consistent with that observed in homogeneous, isotropic turbulence. The Reynolds number scaling of the dissipation moments up to the sixth moment was examined and the results were found to be in good agreement with predicted scaling behaviour (Schumacher et al., Proc. Natl Acad. Sci. USA, vol. 111, 2014, pp. 10961–10965). The probability density functions of the local dissipation scales (Yakhot, Physica D, vol. 215 (2), 2006, pp. 166–174) were also determined and, when the revised local large-scale Reynolds number is used for normalization, provide support for the existence of a universal distribution which scales differently for inner and outer regions.

Self-preservation (SP) analyses are applied to the mean momentum and the scale-by-scale energy budget equations in the far wake of a circular cylinder. The scale-by-scale SP analysis, which is a two-point analysis, complements the SP analysis of the mean momentum equation. Power-law variations are derived for different length scales (e.g. the Taylor microscale and the Kolmogorov length scale) and velocity scales (e.g. the root mean square and the Kolmogorov velocity scale). Further, the SP solutions for the scale-by-scale energy budget equation are exploited to develop an exact relation to estimate the mean turbulent kinetic energy dissipation rate $\bar{{\it\epsilon}}$ on the wake axis. These SP solutions and the new $\bar{{\it\epsilon}}$ relation are well supported by hot-wire data in the far wake at a Reynolds number of 2000 based on the free stream velocity and the cylinder diameter. On the far-wake axis, both the energy spectra and the structure functions exhibit an almost perfect collapse over all wavenumbers and separations, irrespective of the set of scaling variables used for normalisation. This is consistent with a complete self-preservation (i.e. SP is satisfied at all scales of motion) in the far wake.

In this experimental work, the aim is to understand how turbulent thermal flows are enhanced by the destabilization of the boundary layers. Square-stud roughness elements have been added on the bottom plate of a rectangular Rayleigh–Bénard cell in air, to trigger instabilities in the boundary layers. The top plate is kept smooth. The cell proportions are identical to those of the water cell previously operated and described by Salort et al. (Phys. Fluids, vol. 26, 2014, 015112), but six times larger. The very large size of the Barrel of Ilmenau allows detailed velocity fields to be obtained using particle image velocimetry very close to the roughness elements. We found that the flow is quite different at low Rayleigh numbers, where there is no heat-transfer enhancement, and at high Rayleigh numbers where there is a heat-transfer enhancement due to the roughness. Below the transition, the fluid inside the notch, i.e. between the studs, is essentially at rest, though it is slowly recirculating. The velocity profiles on the top of obstacles and in grooves are fairly compatible with those obtained in the smooth case. Above the transition, on the other hand, we observe large incursions of the bulk inside the notch, and the velocity profiles on the top of obstacles are closer to the logarithmic profiles expected in the case of turbulent boundary layers.

We apply a recently developed filtering approach, i.e. filter-space technique (FST), to study the scale-to-scale transport of kinetic energy, thermal energy, and enstrophy in two-dimensional (2D) Rayleigh–Taylor (RT) turbulence. Although the scaling laws of the energy cascades in 2D RT systems follow the Bolgiano–Obukhov (BO59) scenario due to buoyancy forces, the kinetic energy is still found to be, on average, dynamically transferred to large scales by an inverse cascade, while both the mean thermal energy and the mean enstrophy move towards small scales by forward cascades. In particular, there is a reasonably extended range over which the transfer rate of thermal energy is scale-independent and equals the corresponding thermal dissipation rate at different times. This range functions similarly to the inertial range for the kinetic energy in the homogeneous and isotropic turbulence. Our results further show that at small scales the fluctuations of the three instantaneous local fluxes are highly asymmetrically distributed and there is a strong correlation between any two fluxes. These small-scale features are signatures of the mixing and dissipation of fluids with steep temperature gradients at the fluid interfaces.

Structural properties of regions of uniform streamwise momentum in turbulent boundary layers are examined using experimental databases obtained from particle image velocimetry. This investigation employs a large range of Reynolds numbers, spanning more than an order of magnitude ($Re_{{\it\tau}}=10^{3}{-}10^{4}$), enabling us to provide a detailed description of uniform momentum zones as a function of Reynolds number. Our analysis starts by examining the identification criterion used by Adrian et al. (J. Fluid Mech., vol. 422, 2000, pp. 1–54) to report the presence of uniform momentum zones in turbulent boundary layers. This criterion is then applied to show that a zonal-like structural arrangement is prevalent in all datasets examined, emphasising its importance in the structural organisation. Streamwise velocity fluctuations within the zones are observed to be small but they are bounded by distinct step changes in streamwise momentum which indicate that shear layers of intense vorticity separate each zone. A log-linear increase in the number of these zones with increasing Reynolds number is revealed, together with an increase in the thicknesses of zones with increasing distance from the wall. These results support a hierarchical length-scale distribution of coherent structures, which generate zonal-like organisation within turbulent boundary layers. Interpretation of these findings is aided by employing synthetic velocity fields generated using a model based on the attached eddy hypothesis, which is described in the work of Perry and co-workers. Comparisons between the model and experimental results show that a hierarchy of self-similar structures leads to population densities and length-scale distributions of uniform momentum zones that closely adhere to those observed experimentally in this study.

We present a theoretical study of the dispersion of particles along quasi-horizontal stratification surfaces induced by small-amplitude internal gravity waves that are forced by white noise and dissipated by linear or nonlinear damping. This extends previous studies in which only linear damping was considered. The damping itself is a toy model for the nonlinear processes that would attenuate a wave mode in a broad spectrum of internal waves such as the Garrett–Munk spectrum for ocean internal waves. We compute the velocity covariance using an eigenfunction expansion of the Kolmogorov backward equation and investigate how the degree of nonlinearity affects the scaling of diffusivity with wave amplitude. We find a simple new expression for the diffusivity that is valid in both the linear and nonlinear cases, and we consider the likely quantitative importance of these process in the context of data from field experiments on small-scale ocean tracer dispersion.

The onset of monami – the synchronous waving of seagrass beds driven by a steady flow – is modelled as a linear instability of the flow. Unlike previous works, our model considers the drag exerted by the grass in establishing the steady flow profile, and in damping out perturbations to it. We find two distinct modes of instability, which we label modes 1 and 2. Mode 1 is closely related to Kelvin–Helmholtz instability modified by vegetation drag, whereas mode 2 is unrelated to Kelvin–Helmholtz instability and arises from an interaction between the flow in the vegetated and unvegetated layers. The vegetation damping, according to our model, leads to a finite threshold flow for both of these modes. Experimental observations for the onset and frequency of waving compare well with model predictions for the instability onset criteria and the imaginary part of the complex growth rate respectively, but experiments lie in a parameter regime where the two modes can not be distinguished.

Prior modal stability analysis (Kojima et al., Phys. Fluids, vol. 27, 1984, pp. 19–32) predicted that a rising or sedimenting droplet in a viscous fluid is stable in the presence of surface tension no matter how small, in contrast to experimental and numerical results. By performing a non-modal stability analysis, we demonstrate the potential for transient growth of the interfacial energy of a rising droplet in the limit of inertialess Stokes equations. The predicted critical capillary numbers for transient growth agree well with those for unstable shape evolution of droplets found in the direct numerical simulations of Koh & Leal (Phys. Fluids, vol. 1, 1989, pp. 1309–1313). Boundary integral simulations are used to delineate the critical amplitude of the most destabilizing perturbations. The critical amplitude is negatively correlated with the linear optimal energy growth, implying that the transient growth is responsible for reducing the necessary perturbation amplitude required to escape the basin of attraction of the spherical solution.

Suspended particles migrate towards inertial focusing positions close to walls and align into trains in finite inertia conduit flow. The relative contribution of inertial and viscous forces at the particle length scale, defined by the particle Reynolds number ($\mathit{Re}_{p}$), is a key parameter, where $\mathit{Re}_{p}=\langle \dot{\unicode[STIX]{x1D6FE}}\rangle D^{2}/\unicode[STIX]{x1D708}$ depends on the mean shear rate $\langle \dot{\unicode[STIX]{x1D6FE}}\rangle$, particle diameter $D$ and fluid kinematic viscosity $\unicode[STIX]{x1D708}$. Controlling the location of inertial focusing positions and the interparticle distance is critical in applications such as flow cytometry, imaging and cell entrapment in droplets. By using experimental observations in rectangular microchannels and lattice Boltzmann numerical simulations of dilute suspension flow, the spacing between particles aligned in trains is measured. From the modes of the probability density function of interparticle spacing, preferred spacings at $5D$ and $2.5D$ are observed. At lower $\mathit{Re}_{p}$, the preferred spacing forms around $5D$, and with increasing $\mathit{Re}_{p}$ the spacing at $2.5D$ becomes more pronounced. With increasing concentration of the suspension the spacing is influenced by particle crowding effects until stable trains are no longer observed.