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Journal of Fluid Mechanics
Journal Prestige (SJR): 1.591
Citation Impact (citeScore): 3
Number of Followers: 157  
  Hybrid Journal Hybrid journal (It can contain Open Access articles)
ISSN (Print) 0022-1120 - ISSN (Online) 1469-7645
Published by Cambridge University Press Homepage  [372 journals]
  • Nozzles, turbulence, and jet noise prediction
    • Authors: Jonathan B. Freund
      Pages: 1 - 4
      Abstract: Jet noise prediction is notoriously challenging because only subtle features of the flow turbulence radiate sound. The article by Brès et al. (J. Fluid Mech., vol. 851, 2018, pp. 83–124) shows that a well-constructed modelling procedure for the nozzle turbulence can provide unprecedented sub-dB prediction accuracy with modest-scale large-eddy simulations, as confirmed by detailed comparison with turbulence and sound-field measurements. This both illuminates the essential mechanisms of the flow and facilitates prediction for engineering design.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.823
      Issue No: Vol. 860 (2019)
  • Temporal stability analysis of jets of lobed geometry
    • Authors: Benshuai Lyu; Ann P. Dowling
      Pages: 5 - 39
      Abstract: A two-dimensional temporal incompressible stability analysis is performed for lobed jets. The jet base flow is assumed to be parallel and of a vortex-sheet type. The eigenfunctions of this simplified stability problem are expanded using the eigenfunctions of a round jet. The original problem is then formulated as an innovative matrix eigenvalue problem, which can be solved in a very robust and efficient manner. The results show that the lobed geometry changes both the convection velocity and temporal growth rate of the instability waves. However, different modes are affected differently. In particular, mode 0 is not sensitive to the geometry changes, whereas modes of higher orders can be changed significantly. The changes become more pronounced as the number of lobes $N$ and the penetration ratio $\unicode[STIX]{x1D716}$ increase. Moreover, the lobed geometry can cause a previously degenerate eigenvalue ( $\unicode[STIX]{x1D706}_{n}=\unicode[STIX]{x1D706}_{-n}$ ) to become non-degenerate ( $\unicode[STIX]{x1D706}_{n}\neq \unicode[STIX]{x1D706}_{-n}$ ) and lead to opposite changes to the stability characteristics of the corresponding symmetric ( $n$ ) and antisymmetric ( $-n$ ) modes. It is also shown that each eigenmode changes its shape in response to the lobes of the vortex sheet, and the degeneracy of an eigenvalue occurs when the vortex sheet has more symmetric planes than the corresponding mode shape (including both symmetric and antisymmetric planes). The new approach developed in this paper can be used to study the stability characteristics of jets of other arbitrary geometries in a robust and efficient manner.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.865
      Issue No: Vol. 860 (2019)
  • Optimal disturbances and large-scale energetic motions in turbulent
           boundary layers
    • Authors: Timothy B. Davis; Ali Uzun, Farrukh S. Alvi
      Pages: 40 - 80
      Abstract: We examine disturbances leading to optimal energy growth in a spatially developing, zero-pressure-gradient turbulent boundary layer. The slow development of the turbulent mean flow in the streamwise direction is modelled through a parabolized formulation to enable a spatial marching procedure. In the present framework, conventional spatial optimal disturbances arise naturally as the homogeneous solution to the linearized equations subject to a turbulent forcing at particular wavenumber combinations. A wave-like decomposition for the disturbance is considered to incorporate both conventional stationary modes as well as propagating modes formed by non-zero frequency/streamwise wavenumber and representative of convective structures naturally observed in wall turbulence. The optimal streamwise wavenumber, which varies with the spatial development of the turbulent mean flow, is computed locally via an auxiliary optimization constraint. The present approach can then be considered, in part, as an extension of the resolvent-based analyses for slowly developing flows. Optimization results reveal highly amplified disturbances for both stationary and propagating modes. Stationary modes identify peak amplification of structures residing near the centre of the logarithmic layer of the turbulent mean flow. Inner-scaled disturbances reminiscent of near wall streaks, and amplified over short streamwise distances, are identified in the computed streamwise energy spectra. In all cases, however, propagating modes surpass their stationary counterpart in both energy amplification and relative contribution to total fluctuation energy. We identify two classes of large-scale energetic modes associated with the logarithmic and wake layers of the turbulent mean flow. The outer-scaled wake modes agree well with the large-scale motions that populate the wake layer. For high Reynolds numbers, the log modes increasingly dominate the energy spectra with the predicted streamwise and wall-normal scales in agreement with superstructures observed in turbulent boundary layers.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.807
      Issue No: Vol. 860 (2019)
  • A novel non-reflecting boundary condition for fluid dynamics solved by
           smoothed particle hydrodynamics
    • Authors: Pingping Wang; A-Man Zhang, Furen Ming, Pengnan Sun, Han Cheng
      Pages: 81 - 114
      Abstract: Non-reflecting boundary conditions (NRBCs) play an important role in computational fluid dynamics (CFD). A novel NRBC based on the method of characteristics using timeline interpolations is proposed for fluid dynamics solved by smoothed particle hydrodynamics (SPH). It is performed by four layers of particles whose pressures and velocities are obtained through the Lagrange interpolation in the time domain which is derived from the propagation of characteristic waves between particles. The proposed NRBC can allow outward travelling pressure and velocity messages to pass through the boundary without obvious reflection. That is, with the implementation of the NRBC, the solution in a finite computational domain of interest is close to that in an infinite domain. Several numerical tests show that this NRBC is robust and applicable for a broad variety of hydrodynamics ranging from low to high speed.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.852
      Issue No: Vol. 860 (2019)
  • Deformation of a biconcave-discoid capsule in extensional flow and
           electric field
    • Authors: Sudip Das; Shivraj D. Deshmukh, Rochish M. Thaokar
      Pages: 115 - 144
      Abstract: Natural (red blood cells) and artificial biconcave-discoid-shaped capsules have immense biological (a cellular component of blood) and technological (as drug carrier) relevance, respectively. Their low reduced volume allows significant shape changes under external fields such as extensional flows (encountered at junctions and size-varying capillaries in biological flows) and electric fields (in applications such as electroporation and dielectrophoresis). This work demonstrates biconcave-discoid to capped-cylindrical and prolate-spheroid shape transitions of a capsule in uniaxial extensional flow as well as in DC and AC electric fields. The shape changes of a stress-free biconcave-discoid capsule in external fields are important in determining the momentum and mass transfer between the capsule and the medium fluid as well as dielectrophoresis and electroporation phenomena of a capsule in an electric field. The biconcave-discoid to capped-cylindrical/prolate-spheroid shape transition is demonstrated for both a capsule (with parameters relevant to drug delivery) as well as for a red blood cell (physiological conditions). However, significant differences are observed in this shape transition depending upon the applied external fields. In an extensional flow, the pressure-driven transition shows the equator being squeezed in and the poles being pulled out to deform into a capped cylinder at low capillary number and a prolate spheroid at high capillary number. On the other hand, in the transition driven by electric fields, the shoulders of the capsule seem to play a significant role in the dynamics. The shape transition in the electric fields depends upon the relative magnitude of the electric and the hydrodynamic response times, particularly relevant for the dynamics of red blood cells in physiological conditions. A new method of analysing the shape transition of red blood cells in AC electric fields is suggested, where a large separation of time scales is observed between the hydrodynamic and electric responses.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.879
      Issue No: Vol. 860 (2019)
  • Instability of sheared density interfaces
    • Authors: T. S. Eaves; N. J. Balmforth
      Pages: 145 - 171
      Abstract: Of the canonical flow instabilities (Kelvin–Helmholtz, Holmboe-wave and Taylor–Caulfield) of stratified shear flow, the Taylor–Caulfield instability (TCI) has received relatively little attention, and forms the focus of the current study. First, a diagnostic of the linear instability dynamics is developed that exploits the net pseudomomentum to distinguish TCI from the other two instabilities for any given flow profile. Second, the nonlinear dynamics of TCI is studied across its range of unstable horizontal wavenumbers and bulk Richardson numbers using numerical simulation. At small bulk Richardson numbers, a cascade of billow structures of sequentially smaller size may form. For large bulk Richardson numbers, the primary nonlinear travelling waves formed by the linear instability break down via a small-scale, Kelvin–Helmholtz-like roll-up mechanism with an associated large amount of mixing. In all cases, secondary parasitic nonlinear Holmboe waves appear at late times for high Prandtl number. Third, a nonlinear diagnostic is proposed to distinguish between the saturated states of the three canonical instabilities based on their distinctive density–streamfunction and generalised vorticity–streamfunction relations.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.827
      Issue No: Vol. 860 (2019)
  • Viscous growth and rebound of a bubble near a rigid surface
    • Authors: Sébastien Michelin; Giacomo Gallino, François Gallaire, Eric Lauga
      Pages: 172 - 199
      Abstract: Motivated by the dynamics of microbubbles near catalytic surfaces in bubble-powered microrockets, we consider theoretically the growth of a free spherical bubble near a flat no-slip surface in a Stokes flow. The flow at the bubble surface is characterised by a constant slip length allowing us to tune the hydrodynamic mobility of its surface and tackle in one formulation both clean and contaminated bubbles as well as rigid shells. Starting with a bubble of infinitesimal size, the fluid flow and hydrodynamic forces on the growing bubble are obtained analytically. We demonstrate that, depending on the value of the bubble slip length relative to the initial distance to the wall, the bubble will either monotonically drain the fluid separating it from the wall, which will exponentially thin, or it will bounce off the surface once before eventually draining the thin film. Clean bubbles are shown to be a singular limit which always monotonically get repelled from the surface. The bouncing events for bubbles with finite slip lengths are further analysed in detail in the lubrication limit. In particular, we identify the origin of the reversal of the hydrodynamic force direction as due to the change in the flow pattern in the film between the bubble and the surface and to the associated lubrication pressure. Last, the final drainage dynamics of the film is observed to follow a universal algebraic scaling for all finite slip lengths.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.876
      Issue No: Vol. 860 (2019)
  • Passive flight in density-stratified fluids
    • Authors: Try Lam; Lionel Vincent, Eva Kanso
      Pages: 200 - 223
      Abstract: Leaves falling in air and marine larvae settling in water are examples of unsteady descents due to complex interactions between gravitational and aerodynamic forces. Understanding passive flight is relevant to many branches of engineering and science, ranging from estimating the behaviour of re-entry space vehicles to analysing the biomechanics of seed dispersion. The motion of regularly shaped objects falling freely in homogenous fluids is relatively well understood. However, less is known about how density stratification of the fluid medium affects passive flight. In this paper, we experimentally investigate the descent of heavy discs in stably stratified fluids for Froude numbers of order 1 and Reynolds numbers of order 1000. We specifically consider fluttering descents, where the disc oscillates as it falls. In comparison with pure water and homogeneous saltwater fluid, we find that density stratification significantly enhances the radial dispersion of the disc, while simultaneously decreasing the vertical descent speed, fluttering amplitude and inclination angle of the disc during descent. We explain the physical mechanisms underlying these observations in the context of a quasi-steady force and torque model. These findings could have significant impact on the design of unpowered vehicles and on the understanding of geological and biological transport where density and temperature variations may occur.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.862
      Issue No: Vol. 860 (2019)
  • Slug generation processes in co-current turbulent-gas/laminar-liquid flows
           in horizontal channels
    • Authors: Sha Miao; Kelli Hendrickson, Yuming Liu
      Pages: 224 - 257
      Abstract: We theoretically and computationally investigate the physical processes of slug-flow development in concurrent two-phase turbulent-gas/laminar-liquid flows in horizontal channels. The objective is to understand the fundamental mechanisms governing the initial growth and subsequent nonlinear evolution of interfacial waves, starting from a smooth stratified flow of two fluids with disparity in density and viscosity and ultimately leading to the formation of intermittent slug flow. We numerically simulate the entire slug development by means of a fully coupled immersed flow (FCIF) solver that couples the two disparate flow dynamics through an immersed boundary (IB) method. From the analysis of spatial/temporal interface evolution, we find that slugs develop through three major cascading processes: (I) stratified-to-wavy transition; (II) development and coalescence of long solitary waves; and (III) rapid channel bridging leading to slugging. In Process I, relatively short interfacial waves form on the smooth interface, whose growth is governed by the Orr–Sommerfeld instability. In Process II, interfacial waves evolve into long solitary waves through multiple resonant and near-resonant wave–wave interactions. From instability analysis of periodic solitary waves, we show that these waves are unstable to their subharmonic disturbances and grow in amplitude and primary wavelength through wave coalescence. The interfacial forcing from the turbulent gas–laminar liquid interactions significantly precipitates the growth of instability of solitary waves and enhances coalescence of solitary waves. In Process III, we show by an asymptotic analysis that interfacial waves achieve multiple-exponential growth right before bridging the channel, consistent with observations in existing experiments. The present study provides important insights for effective modelling of slug-flow dynamics and the prediction of slug frequency and length, important for design and operation of (heavy-oil/gas) pipelines and production facilities.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.868
      Issue No: Vol. 860 (2019)
  • Turbulent flow through a high aspect ratio cooling duct with asymmetric
           wall heating
    • Authors: Thomas Kaller; Vito Pasquariello, Stefan Hickel, Nikolaus A. Adams
      Pages: 258 - 299
      Abstract: We present well-resolved large-eddy simulations of turbulent flow through a straight, high aspect ratio cooling duct operated with water at a bulk Reynolds number of $Re_{b}=110\times 10^{3}$ and an average Nusselt number of $Nu_{xz}=371$ . The geometry and boundary conditions follow an experimental reference case and good agreement with the experimental results is achieved. The current investigation focuses on the influence of asymmetric wall heating on the duct flow field, specifically on the interaction of turbulence-induced secondary flow and turbulent heat transfer, and the associated spatial development of the thermal boundary layer and the inferred viscosity variation. The viscosity reduction towards the heated wall causes a decrease in turbulent mixing, turbulent length scales and turbulence anisotropy as well as a weakening of turbulent ejections. Overall, the secondary flow strength becomes increasingly less intense along the length of the spatially resolved heated duct as compared to an adiabatic duct. Furthermore, we show that the assumption of a constant turbulent Prandtl number is invalid for turbulent heat transfer in an asymmetrically heated duct.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.836
      Issue No: Vol. 860 (2019)
  • Fundamental equations for primary fluid recovery from porous media
    • Authors: Yan Jin; Kang Ping Chen
      Pages: 300 - 317
      Abstract: Primary fluid recovery from a porous medium is driven by the volumetric expansion of the in situ fluid. For production from a petroleum reservoir, primary recovery accounts for more than half of the total amount of recovered hydrocarbon. The primary recovery process is studied here at the pore scale and the macroscopic scale. The pore-scale flow is first analysed using the compressible Navier–Stokes equations and the mathematical theory for low-Mach-number flow developed by Klainerman & Majda (Commun. Pure Appl. Maths, vol. 34 (4), 1981, pp. 481–524; vol. 35 (5), 1982, pp. 629–651). An asymptotic analysis shows that the pore-scale flow is governed by the self-diffusion of the fluid and it exhibits a slip-like mass flow rate, even though the velocity satisfies the no-slip condition on the pore wall. The pore-scale density equation is then upscaled to a macroscopic diffusion equation for the density which possesses a diffusion coefficient proportional to the fluid’s kinematic viscosity. Darcy’s law is shown to be inapplicable to primary fluid recovery and it should be replaced by a new mass flux equation which depends on the porosity but not on the permeability. This is in stark contrast to the classical result and it can have important implications for hydrocarbon recovery as well as other applications.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.874
      Issue No: Vol. 860 (2019)
  • Computationally generated constitutive models for particle phase rheology
           in gas-fluidized suspensions
    • Authors: Yile Gu; Ali Ozel, Jari Kolehmainen, Sankaran Sundaresan
      Pages: 318 - 349
      Abstract: Developing constitutive models for particle phase rheology in gas-fluidized suspensions through rigorous statistical mechanical methods is very difficult when complex inter-particle forces are present. In the present study, we pursue a computational approach based on results obtained through Eulerian–Lagrangian simulations of the fluidized state. Simulations were performed in a periodic domain for non-cohesive and mildly cohesive (Geldart Group A) particles. Based on the simulation results, we propose modified closures for pressure, bulk viscosity, shear viscosity and the rate of dissipation of pseudo-thermal energy. For non-cohesive particles, results in the high granular temperature $T$ regime agree well with constitutive expressions afforded by the kinetic theory of granular materials, demonstrating the validity of the methodology. The simulations reveal a low $T$ regime, where the inter-particle collision time is determined by gravitational fall between collisions. Inter-particle cohesion has little effect in the high $T$ regime, but changes the behaviour appreciably in the low $T$ regime. At a given $T$ , a cohesive particle system manifests a lower pressure at low particle volume fractions when compared to non-cohesive systems; at higher volume fractions, the cohesive assemblies attain higher coordination numbers than the non-cohesive systems, and experience greater pressures. Cohesive systems exhibit yield stress, which is weakened by particle agitation, as characterized by $T$ . All these effects are captured through simple modifications to the kinetic theory of granular materials for non-cohesive materials.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.856
      Issue No: Vol. 860 (2019)
  • Turbulence in intermittent transitional boundary layers and in turbulence
    • Authors: Olaf Marxen; Tamer A. Zaki
      Pages: 350 - 383
      Abstract: Direct numerical simulation data of bypass transition in flat-plate boundary layers are analysed to examine the characteristics of turbulence in the transitional regime. When intermittency is 50 % or less, the flow features a juxtaposition of turbulence spots surrounded by streaky laminar regions. Conditionally averaged turbulence statistics are evaluated within the spots, and are compared to standard time averaging in both the transition region and in fully turbulent boundary layers. The turbulent-conditioned root-mean-square levels of the streamwise velocity perturbations are notably elevated in the early transitional boundary layer, while the wall-normal and spanwise components are closer to the levels typical for fully turbulent flow. The analysis is also extended to include ensemble averaging of the spots. When the patches of turbulence are sufficiently large, they develop a core region with similar statistics to fully turbulent boundary layers. Within the tip and the wings of the spots, however, the Reynolds stresses and terms in the turbulence kinetic energy budget are elevated. The enhanced turbulence production in the transition zone, which exceeds the levels from fully turbulent boundary layers, contributes to the higher skin-friction coefficient in that region. Qualitatively, the same observations hold for different spot sizes and levels of free-stream turbulence, except for young spots which do not yet have a core region of developed turbulence.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.822
      Issue No: Vol. 860 (2019)
  • The effect of vertically varying permeability on tracer dispersion
    • Authors: Edward M. Hinton; Andrew W. Woods
      Pages: 384 - 407
      Abstract: We study the migration of a tracer within an injection-driven flow in a horizontal aquifer in which the permeability varies with depth. The permeability gradient produces a shear and this leads to lateral dispersion of the tracer. In the high permeability regions, the tracer moves substantially faster than the mean flow and eventually enters the nose region of the flow where the depth of the current is less than the depth of the aquifer. Depending on the influence of (i) the viscosity contrast between the injected fluid and the original fluid, and (ii) the vertical permeability gradient, the nose of the current may be of fixed shape or may gradually lengthen with time. This leads to a variety of patterns of dispersal of the tracer, which may either remain in the nose or cycle through the nose and be left behind. Our results illustrate the complexity of the migration of a tracer in a heterogeneous aquifer which has important implications for interpreting the results of tracer tests as may be proposed for monitoring $\text{CO}_{2}$ or gas injected into subsurface reservoirs.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.891
      Issue No: Vol. 860 (2019)
  • On the dynamics of a free surface of an ideal fluid in a bounded domain in
           the presence of surface tension
    • Authors: Sergey A. Dyachenko
      Pages: 408 - 418
      Abstract: We derive a set of equations in conformal variables that describe a potential flow of an ideal two-dimensional inviscid fluid with free surface in a bounded domain. This formulation is free of numerical instabilities present in the equations for the surface elevation and potential derived in Dyachenko et al. (Plasma Phys. Rep. vol. 22 (10), 1996, pp. 829–840) with some restrictions on analyticity relieved, which allows to treat a finite volume of fluid enclosed by a free-moving boundary. We illustrate with a comparison of numerical simulations of the Dirichlet ellipse, an exact solution for a zero surface tension fluid. We demonstrate how the oscillations of the free surface of a unit disc droplet may lead to breaking of one droplet into two when surface tension is present.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.885
      Issue No: Vol. 860 (2019)
  • Simulation of air–water interfacial mass transfer driven by
           high-intensity isotropic turbulence
    • Authors: H. Herlina; J. G. Wissink
      Pages: 419 - 440
      Abstract: Previous direct numerical simulations (DNS) of mass transfer across the air–water interface have been limited to low-intensity turbulent flow with turbulent Reynolds numbers of $R_{T}\leqslant 500$ . This paper presents the first DNS of low-diffusivity interfacial mass transfer across a clean surface driven by high-intensity ( $1440\leqslant R_{T}\leqslant 1856$ ) isotropic turbulent flow diffusing from below. The detailed results, presented here for Schmidt numbers $Sc=20$ and $500$ , support the validity of theoretical scaling laws and existing experimental data obtained at high $R_{T}$ . In the DNS, to properly resolve the turbulent flow and the scalar transport at $Sc=20$ , up to $524\times 10^{6}$ grid points were needed, while $65.5\times 10^{9}$ grid points were required to resolve the scalar transport at $Sc=500$ , which is typical for oxygen in water. Compared to the low- $R_{T}$ simulations, where turbulent mass flux is dominated by large eddies, in the present high-
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.884
      Issue No: Vol. 860 (2019)
  • Structure of the hydraulic jump in convergent radial flows
    • Authors: K. A. Ivanova; S. L. Gavrilyuk
      Pages: 441 - 464
      Abstract: We are interested in the modelling of multi-dimensional turbulent hydraulic jumps in convergent radial flow. To describe the formation of intensive eddies (rollers) at the front of the hydraulic jump, a new model of shear shallow water flows is used. The governing equations form a non-conservative hyperbolic system with dissipative source terms. The structure of equations is reminiscent of generic Reynolds-averaged Euler equations for barotropic compressible turbulent flows. Two types of dissipative term are studied. The first one corresponds to a Chézy-like dissipation rate, and the second one to a standard energy dissipation rate commonly used in compressible turbulence. Both of them guarantee the positive definiteness of the Reynolds stress tensor. The equations are rewritten in polar coordinates and numerically solved by using an original splitting procedure. Numerical results for both types of dissipation are presented and qualitatively compared with the experimental works. The results show both experimentally observed phenomena (cusp formation at the front of the hydraulic jump) as well as new flow patterns (the shape of the hydraulic jump becomes a rotating square).
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.901
      Issue No: Vol. 860 (2019)
  • Scale-dependent alignment, tumbling and stretching of slender rods in
           isotropic turbulence
    • Authors: Nimish Pujara; Greg A. Voth, Evan A. Variano
      Pages: 465 - 486
      Abstract: We examine the dynamics of slender, rigid rods in direct numerical simulation of isotropic turbulence. The focus is on the statistics of three quantities and how they vary as rod length increases from the dissipation range to the inertial range. These quantities are (i) the steady-state rod alignment with respect to the perceived velocity gradients in the surrounding flow, (ii) the rate of rod reorientation (tumbling) and (iii) the rate at which the rod end points move apart (stretching). Under the approximations of slender-body theory, the rod inertia is neglected and rods are modelled as passive particles in the flow that do not affect the fluid velocity field. We find that the average rod alignment changes qualitatively as rod length increases from the dissipation range to the inertial range. While rods in the dissipation range align most strongly with fluid vorticity, rods in the inertial range align most strongly with the most extensional eigenvector of the perceived strain-rate tensor. For rods in the inertial range, we find that the variance of rod stretching and the variance of rod tumbling both scale as $l^{-4/3}$ , where $l$ is the rod length. However, when rod dynamics are compared to two-point fluid velocity statistics (structure functions), we see non-monotonic behaviour in the variance of rod tumbling due to the influence of small-scale fluid motions. Additionally, we find that the skewness of rod stretching does not show scale invariance in the inertial range, in contrast to the skewness of longitudinal fluid velocity increments as predicted by Kolmogorov’s $4/5$ law. Finally, we examine the power-law scaling exponents of higher-order moments of rod tumbling and rod stretching for rods with lengths in the inertial range and find that they show anomalous scaling. We compare these scaling exponents to predictions from Kolmogorov’s refined similarity hypotheses.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.866
      Issue No: Vol. 860 (2019)
  • The enhancement of viscous fingering with bidisperse particle suspension
    • Authors: Feng Xu; Sungyon Lee
      Pages: 487 - 509
      Abstract: Viscous fingering is observed experimentally when a bidisperse suspension displaces air inside a Hele-Shaw cell, despite the stabilising viscosity ratio between the invading (suspension) and defending (air) phases. Careful experiments are carried out to characterise this instability by either systematically varying the large-particle concentrations $\unicode[STIX]{x1D719}_{l0}$ at constant total concentrations $\unicode[STIX]{x1D719}_{0}$ , or changing $\unicode[STIX]{x1D719}_{0}$ with fixed $\unicode[STIX]{x1D719}_{l0}$ . Leading to the instability, we observe that larger particles consistently enrich the fluid–fluid interface at a faster rate than small particles. This size-dependent enrichment of the interface leads to an earlier onset of the fingering instability for bidisperse suspensions, compared to their monodisperse counterpart of all small particles. In particular, even the small presence of large particles is shown to effectively lower the total particle concentration needed for fingering, compared to the all-small-particle case. We hypothesise that the key mechanism behind this enhanced viscous fingering is the size-dependent nature of shear-induced migration of particles far upstream from the interface. A reduced equilibrium model is derived based on the modified suspension balance model to verify this hypothesis, in reasonable agreement with experiments.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.846
      Issue No: Vol. 860 (2019)
  • Dynamical similarity and universality of drop size and velocity spectra in
    • Authors: K. Dhivyaraja; D. Gaddes, E. Freeman, S. Tadigadapa, M. V. Panchagnula
      Pages: 510 - 543
      Abstract: Sprays are a class of multiphase flows which exhibit a wide range of drop size and velocity scales spanning several orders of magnitude. The objective of the current work is to experimentally investigate the prospect of dynamical similarity in these flows. We are also motivated to identify a choice of length and time scales which could lead towards a universal description of the drop size and velocity spectra. Towards this end, we have fabricated a cohort of geometrically similar pressure swirl atomizers using micro-electromechanical systems (MEMS) as well as additive manufacturing technology. We have characterized the dynamical characteristics of the sprays as well as the drop size and velocity spectra (in terms of probability density functions, p.d.f.s) over a wide range of Reynolds ( $Re$ ) and Weber numbers ( $We$ ) using high-speed imaging and phase Doppler interferometry, respectively. We show that the dimensionless Sauter mean diameter ( $D_{32}$ ) scaled to the boundary layer thickness in the liquid sheet at the nozzle exit ( $\unicode[STIX]{x1D6FF}_{o}$ ) exhibits self-similarity in the core region of the spray, but not in the outer zone. In addition, we show that global drop size spectra in the sprays show two distinct characteristics. The spectra from varying $Re$ and $We$ collapse onto a universal p.d.f. for drops of size $x$ where $x/\unicode[STIX]{x1D6FF}_{o}>1$ . For $x/\unicode[STIX]{x1D6FF}_{o}
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.893
      Issue No: Vol. 860 (2019)
  • Radiometric flow in periodically patterned channels: fluid physics and
           improved configurations
    • Authors: Ali Lotfian; Ehsan Roohi
      Pages: 544 - 576
      Abstract: With the aid of direct simulation Monte Carlo (DSMC), we conduct a detailed investigation pertaining to the fluid and thermal characteristics of rarefied gas flow with regard to various arrangements for radiometric pumps featuring vane and ratchet structures. For the same, we consider three categories of radiometric pumps consisting of channels with their bottom or top surfaces periodically patterned with different structures. The structures in the design of the first category are assumed to be on the bottom wall and consist of either a simple vane, a right-angled triangular fin or an isosceles triangular fin. The arrangements on the second category of radiometric pumps consist of an alternating diffuse–specular right-angled fin and an alternating diffuse–specular isosceles fin on the bottom wall. The third category contains either a channel with double isosceles triangular fins on its lowermost surface or a zigzag channel with double isosceles triangular fins on both walls. In the first and the third categories, the surfaces of the channel and its structures are considered as diffuse reflectors. The temperature is kept steady on the horizontal walls of the channel; thus, radiometric flow is created by subjecting the adjacent sides of the vane/ratchet to constant but unequal temperatures. On the other hand, for the second category, radiometric flow is introduced by specifying different top/bottom channel wall temperatures. The DSMC simulations are performed at a Knudsen number based on the vane/ratchet height of approximately one. The dominant mechanism in the radiometric force production is clarified for the examined configurations. Our results demonstrate that, at the investigated Knudsen number, the zigzag channel experiences maximum induced velocity with a parabolic velocity profile, whereas its net radiometric force vanishes. In the case of all other configurations, the flow pattern resembles a Couette flow in the open section of the channel situated above the vane/ratchet. To further enhance the simulations, the predictions of the finite volume discretization of the Boltzmann Bhatnagar–Gross–Krook (BGK)–Shakhov equation for the mass flux dependence on temperature difference and Knudsen number are also reported for typical test cases.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.880
      Issue No: Vol. 860 (2019)
  • Barotropic theory for the velocity profile of Jupiter’s turbulent jets:
           an example for an exact turbulent closure
    • Authors: E. Woillez; F. Bouchet
      Pages: 577 - 607
      Abstract: We model the dynamics of Jupiter’s jets by the stochastic barotropic $\unicode[STIX]{x1D6FD}$ -plane model. In this simple framework, by analytic computation of the averaged effect of eddies, we obtain three new explicit results about the equilibrium structure of jets. First we obtain a very simple explicit relation between the Reynolds stresses, the energy injection rate and the averaged velocity shear. This predicts the averaged velocity profile far from the jet edges (extrema of zonal velocity). Our approach takes advantage of a time-scale separation between the inertial dynamics on one hand, and the spin-up (or spin-down) time on the other. Second, a specific asymptotic expansion close to the eastward jet extremum explains the formation of a cusp at the scale of energy injection, characterized by a curvature that is independent of the forcing spectrum. Finally, we derive equations that describe the evolution of the westward tip of the jets. The analysis of these equations is consistent with the previously discussed picture of barotropic adjustment, explaining the relation between the westward jet curvature and the $\unicode[STIX]{x1D6FD}$ -effect. Our results give a consistent overall theory of the stationary velocity profile of inertial barotropic zonal jets, in the limit of small-scale forcing.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.877
      Issue No: Vol. 860 (2019)
  • Suppression of the Kapitza instability in confined falling liquid films
    • Authors: Gianluca Lavalle; Yiqin Li, Sophie Mergui, Nicolas Grenier, Georg F. Dietze
      Pages: 608 - 639
      Abstract: We revisit the linear stability of a falling liquid film flowing through an inclined narrow channel in interaction with a gas phase. We focus on a particular region of parameter space, small inclination and very strong confinement, where we have found the gas to strongly stabilize the film, up to the point of fully suppressing the long-wave interfacial instability attributed to Kapitza (Zh. Eksp. Teor. Fiz., vol. 18 (1), 1948, pp. 3–28). The stabilization occurs both when the gas is merely subject to an aerostatic pressure difference, i.e. when the pressure difference balances the weight of the gas column, and when it flows counter-currently. In the latter case, the degree of stabilization increases with the gas velocity. Our investigation is based on a numerical solution of the Orr–Sommerfeld temporal linear stability problem as well as stability experiments that clearly confirm the observed effect. We quantify the degree of stabilization by comparing the linear stability threshold with its passive-gas limit, and perform a parametric study, varying the relative confinement, the Reynolds number, the inclination angle and the Kapitza number. For example, we find a 30 % reduction of the cutoff wavenumber of the instability for a water film in contact with air, flowing through a channel inclined at $3^{\circ }$ and of height 2.8 times the film thickness. We also identify the critical conditions for the full suppression of the instability in terms of the governing parameters. The stabilization is caused by the strong confinement of the gas, which produces perturbations of the adverse interfacial tangential shear stress that are shifted by half a wavelength with respect to the wavy film surface. This tends to reduce flow-rate variations within the film, thus attenuating the inertia-based driving mechanism of the Kapitza instability.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.902
      Issue No: Vol. 860 (2019)
  • A fate-alternating transitional regime in contracting liquid filaments
    • Authors: F. Wang; F. P. Contò, N. Naz, J. R. Castrejón-Pita, A. A. Castrejón-Pita, C. G. Bailey, W. Wang, J. J. Feng, Y. Sui
      Pages: 640 - 653
      Abstract: The fate of a contracting liquid filament depends on the Ohnesorge number ( $Oh$ ), the initial aspect ratio ( $\unicode[STIX]{x1D6E4}$ ) and surface perturbation. Generally, it is believed that there exists a critical aspect ratio $\unicode[STIX]{x1D6E4}_{c}(Oh)$ such that longer filaments break up and shorter ones recoil into a single drop. Through computational and experimental studies, we report a transitional regime for filaments with a broad range of intermediate aspect ratios, where there exist multiple $\unicode[STIX]{x1D6E4}_{c}$ thresholds at which a novel breakup mode alternates with no-break mode. We develop a simple model considering the superposition of capillary waves, which can predict the complicated new phase diagram. In this model, the breakup results from constructive interference between the capillary waves that originate from the ends of the filament.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.855
      Issue No: Vol. 860 (2019)
  • Non-equilibrium effects on flow past a circular cylinder in the slip and
           early transition regime
    • Authors: Xiao-Jun Gu; Robert W. Barber, Benzi John, David R. Emerson
      Pages: 654 - 681
      Abstract: This paper presents a comprehensive investigation into flow past a circular cylinder where compressibility and rarefaction effects play an important role. The study focuses on steady subsonic flow in the Reynolds-number range 0.1–45. Rarefaction, or non-equilibrium, effects in the slip and early transition regime are accounted for using the method of moments and results are compared to data from kinetic theory obtained from the direct simulation Monte Carlo method. Solutions obtained for incompressible continuum flow serve as a baseline to examine non-equilibrium effects on the flow features. For creeping flow, where the Reynolds number is less than unity, the drag coefficient predicted by the moment equations is in good agreement with kinetic theory for Knudsen numbers less than one. When flow separation occurs, we show that the effects of rarefaction and velocity slip delay flow separation and will reduce the size of the vortices downstream of the cylinder. When the Knudsen number is above 0.028, the vortex length shows an initial increase with the Reynolds number, as observed in the standard no-slip continuum regime. However, once the Reynolds number exceeds a critical value, the size of the downstream vortices decreases with increasing Reynolds number until they disappear. An existence criterion, which identifies the limits for the presence of the vortices, is proposed. The flow physics around the cylinder is further analysed in terms of velocity slip, pressure and skin friction coefficients, which highlights that viscous, rarefaction and compressibility effects all play a complex role. We also show that the local Knudsen number, which indicates the state of the gas around the cylinder, can differ significantly from its free-stream value and it is essential that computational studies of subsonic gas flows in the slip and early transition regime are able to account for these strong non-equilibrium effects.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.869
      Issue No: Vol. 860 (2019)
  • Shear thinning in non-Brownian suspensions explained by variable friction
           between particles
    • Authors: Laurent Lobry; Elisabeth Lemaire, Frédéric Blanc, Stany Gallier, François Peters
      Pages: 682 - 710
      Abstract: We propose to explain shear-thinning behaviour observed in most concentrated non-Brownian suspensions by variable friction between particles. Considering the low magnitude of the forces experienced by the particles of suspensions under shear flow, it is first argued that rough particles come into solid contact through one or a few asperities. In such a few-asperity elastic–plastic contact, the friction coefficient is expected not to be constant but to decrease with increasing normal load. Simulations based on the force coupling method and including such a load-dependent friction coefficient are performed for various particle volume fractions. The results of the numerical simulations are compared to viscosity measurements carried out on suspensions of polystyrene particles ( $40~\unicode[STIX]{x03BC}\text{m}$ in diameter) dispersed in a Newtonian silicon oil. The agreement is shown to be satisfactory. Furthermore, the comparison between the simulations conducted either with a constant or a load-dependent friction coefficient provides a model for the shear-thinning viscosity. In this model the effective friction coefficient $\unicode[STIX]{x1D707}^{eff}$ is specified by the effective normal contact force which is simply proportional to the bulk shear stress. As the shear stress increases, $\unicode[STIX]{x1D707}^{eff}$ decreases and the jamming volume fraction increases, leading to the reduction of the viscosity. Finally, using this model, we show that it is possible to evaluate the microscopic friction coefficient for each applied shear stress from the rheometric measurements.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.881
      Issue No: Vol. 860 (2019)
  • Self-propulsion near the onset of Marangoni instability of deformable
           active droplets
    • Authors: Matvey Morozov; Sébastien Michelin
      Pages: 711 - 738
      Abstract: Experimental observations indicate that chemically active droplets suspended in a surfactant-laden fluid can self-propel spontaneously. The onset of this motion is attributed to a symmetry-breaking Marangoni instability resulting from the nonlinear advective coupling of the distribution of surfactant to the hydrodynamic flow generated by Marangoni stresses at the droplet’s surface. Here, we use a weakly nonlinear analysis to characterize the self-propulsion near the instability threshold and the influence of the droplet’s deformability. We report that, in the vicinity of the threshold, deformability enhances self-propulsion of viscous droplets, but hinders propulsion of drops that are roughly less viscous than the surrounding fluid. Our asymptotics further reveals that droplet deformability may alter the type of bifurcation leading to symmetry breaking: for moderately deformable droplets, the onset of self-propulsion is transcritical and a regime of steady self-propulsion is stable; while in the case of highly deformable drops, no steady flows can be found within the asymptotic limit considered in this paper, suggesting that the bifurcation is subcritical.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.853
      Issue No: Vol. 860 (2019)
  • Flow-induced vibrations of a rotating cylinder in an arbitrary direction
    • Authors: Rémi Bourguet
      Pages: 739 - 766
      Abstract: The flow-induced vibrations of an elastically mounted circular cylinder, free to oscillate in an arbitrary direction and forced to rotate about its axis, are examined via two- and three-dimensional simulations, at a Reynolds number equal to 100, based on the body diameter and inflow velocity. The behaviour of the flow–structure system is investigated over the entire range of vibration directions, defined by the angle $\unicode[STIX]{x1D703}$ between the direction of the current and the direction of motion, a wide range of values of the reduced velocity $U^{\star }$ (inverse of the oscillator natural frequency) and three values of the rotation rate (ratio between the cylinder surface and inflow velocities), $\unicode[STIX]{x1D6FC}\in \{0,1,3\}$ , in order to cover the reference non-rotating cylinder case, as well as typical slow and fast rotation cases. The oscillations of the non-rotating cylinder ( $\unicode[STIX]{x1D6FC}=0$ ) develop under wake-body synchronization or lock-in, and their amplitude exhibits a bell-shaped evolution, typical of vortex-induced vibrations (VIV), as a function of $U^{\star }$ . When $\unicode[STIX]{x1D703}$ is increased from $0^{\circ }$ to $90^{\circ }$ (or decreased from $180^{\circ }$ to
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.896
      Issue No: Vol. 860 (2019)
  • Laboratory recreation of the Draupner wave and the role of breaking in
           crossing seas
    • Authors: M. L. McAllister; S. Draycott, T. A. A. Adcock, P. H. Taylor, T. S. van den Bremer
      Pages: 767 - 786
      Abstract: Freak or rogue waves are so called because of their unexpectedly large size relative to the population of smaller waves in which they occur. The 25.6 m high Draupner wave, observed in a sea state with a significant wave height of 12 m, was one of the first confirmed field measurements of a freak wave. The physical mechanisms that give rise to freak waves such as the Draupner wave are still contentious. Through physical experiments carried out in a circular wave tank, we attempt to recreate the freak wave measured at the Draupner platform and gain an understanding of the directional conditions capable of supporting such a large and steep wave. Herein, we recreate the full scaled crest amplitude and profile of the Draupner wave, including bound set-up. We find that the onset and type of wave breaking play a significant role and differ significantly for crossing and non-crossing waves. Crucially, breaking becomes less crest-amplitude limiting for sufficiently large crossing angles and involves the formation of near-vertical jets. In our experiments, we were only able to reproduce the scaled crest and total wave height of the wave measured at the Draupner platform for conditions where two wave systems cross at a large angle.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.886
      Issue No: Vol. 860 (2019)
  • The effects of stable stratification on the decay of initially isotropic
           homogeneous turbulence
    • Authors: Stephen M. de Bruyn Kops; James J. Riley
      Pages: 787 - 821
      Abstract: We report on direct numerical simulations of the decay of initially isotropic, homogeneous turbulence subject to the application of stable density stratification. Flows were simulated for three different initial Reynolds numbers, but for the same initial Froude number. We find that the flows pass through three different dynamical regimes as they decay, depending on the local values of the Froude number and activity parameter. These regimes are analogous to those seen in the experimental study of Spedding (J. Fluid Mech., vol. 337, 1997, pp. 283–301) for the wake of a sphere. The flows initially decay with little influence of stratification, up to approximately one buoyancy period, when the local Froude number has dropped below 1. At this point the flows have adjusted to the density stratification, and, if the activity parameter is large enough, begin to decay at a slower rate and spread horizontally at a faster rate, consistent with the predictions of Davidson (J. Fluid Mech., vol. 663, 2010, pp. 268–292) and the scaling arguments of Billant & Chomaz (Phys. Fluids, vol. 13, 2001, pp. 1645–1651). We refer to this second regime as the stratified turbulence regime. As the flows continue to decay, ultimately the activity parameter drops below approximately 1 as viscous effects begin to dominate. In this regime, the flows have become quasi-horizontal, and approximately obey the scaling arguments of Godoy-Diana et al. (J. Fluid Mech., vol. 504, 2004, pp. 229–238).
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.888
      Issue No: Vol. 860 (2019)
  • Viscosity, surface tension and gravity effects on acoustic reflection and
    • Authors: R. Krechetnikov
      Pages: 822 - 836
      Abstract: The idea of the present work is to study from a unifying viewpoint the effects of viscosity, surface tension and gravity on acoustic reflection and refraction at a fluid interface, with the focus on modifications of Snell’s (Snell–Descartes’) law. While all these effects can be treated individually due to separation of the associated time scales, the contributions of surface tension to the gravity and viscosity cases are considered as well. The analysis reveals a number of phenomena among which are dispersive refraction laws, surface tension enhancing reflection, acoustic field generating vorticity at the interface, and viscosity enhancing/suppressing reflection as well as giving rise to extra reflected and transmitted waves.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.897
      Issue No: Vol. 860 (2019)
  • Stability of flow through deformable channels and tubes: implications of
           consistent formulation
    • Authors: Ramkarn Patne; V. Shankar
      Pages: 837 - 885
      Abstract: The present study is aimed at assessing the existing results concerning the stability of canonical shear flows in channels and tubes with deformable walls, in light of consistent formulations of the nonlinear solid constitutive model and linearised interface conditions at the fluid–solid interface. We show that a class of unstable shear-wave modes at low Reynolds number, predicted by previous studies for pressure-driven flows through neo-Hookean tubes and channels, is absent upon use of consistent interfacial conditions. Furthermore, we analyse the consequences of the change in solid model on the stability of the canonical shear flows by using both neo-Hookean and Mooney–Rivlin models. We show that the salient features of the stability of the system are adequately captured by a consistent formulation of the neo-Hookean solid model, thus precluding the need to employ more detailed solid models. The stability analysis of planar flows past a neo-Hookean solid subjected to three-dimensional disturbances showed that two-dimensional disturbances are more unstable than the corresponding three-dimensional disturbances within the consistent formulations. We show that prior inconsistent formulations of the solid constitutive equation predict a physically spurious spanwise instability in disagreement with experiments thereby demonstrating their inapplicability to predict instabilities in flow past deformable solid surfaces. Using the consistent formulation, the present work provides an accurate picture, over a range of Reynolds numbers, of the stability of canonical shear flows through deformable channels and tubes. Importantly, it is shown how inconsistencies in either the bulk constitutive relation or in the linearisation of the interface conditions can separately lead to physically spurious instabilities. The predictions of this work are relevant to experimental studies in flow through deformable tubes and channels in the low and moderate Reynolds number regime.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.908
      Issue No: Vol. 860 (2019)
  • Spectral analysis of the budget equation in turbulent channel flows at
           high Reynolds number
    • Authors: Myoungkyu Lee; Robert D. Moser
      Pages: 886 - 938
      Abstract: The transport equations for the variances of the velocity components are investigated using data from direct numerical simulations of incompressible channel flows at friction Reynolds number ( $Re_{\unicode[STIX]{x1D70F}}$ ) up to $Re_{\unicode[STIX]{x1D70F}}=5200$ . Each term in the transport equation has been spectrally decomposed to expose the contribution of turbulence at different length scales to the processes governing the flow of energy in the wall-normal direction, in scale and among components. The outer-layer turbulence is dominated by very large-scale streamwise elongated modes, which are consistent with the very large-scale motions (VLSM) that have been observed by many others. The presence of these VLSMs drives many of the characteristics of the turbulent energy flows. Away from the wall, production occurs primarily in these large-scale streamwise-elongated modes in the streamwise velocity, but dissipation occurs nearly isotropically in both velocity components and scale. For this to happen, the energy is transferred from the streamwise-elongated modes to modes with a range of orientations through nonlinear interactions, and then transferred to other velocity components. This allows energy to be transferred more-or-less isotropically from these large scales to the small scales at which dissipation occurs. The VLSMs also transfer energy to the wall region, resulting in a modulation of the autonomous near-wall dynamics and the observed Reynolds number dependence of the near-wall velocity variances. The near-wall energy flows are more complex, but are consistent with the well-known autonomous near-wall dynamics that gives rise to streaks and streamwise vortices. Through the overlap region between outer- and inner-layer turbulence, there is a self-similar structure to the energy flows. The VLSM production occurs at spanwise scales that grow with $y$ . There is transport of energy away from the wall over a range of scales that grows with $y$ . Moreover, there is transfer of energy to small dissipative scales which grows like $y^{1/4}$ , as expected from Kolmogorov scaling. Finally, the small-scale near-wall processes characterised by wavelengths less than 1000 wall units are largely Reynolds number independent, while the larger-scale outer-layer processes are strongly Reynolds number dependent. The interaction between them appears to be relatively simple.
      PubDate: 2019-02-10T00:00:00.000Z
      DOI: 10.1017/jfm.2018.903
      Issue No: Vol. 860 (2019)
School of Mathematical and Computer Sciences
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