Energy flux plays a key role in analyses of energy-cascading turbulence. In isotropic turbulence, the flux is given by a scalar as a function of the magnitude of the wavenumber. On the other hand, the flux in anisotropic turbulence should be a geometric vector that has a direction as well as a magnitude, and depends not only on the magnitude of the wavenumber but also on its direction. The energy-flux vector in anisotropic turbulence cannot be uniquely determined in a way used for the isotropic flux. In this work, introducing two ansatzes, net locality and efficiency of nonlinear energy transfer, we propose a way to determine the energy-flux vector in anisotropic turbulence using the Moore–Penrose inverse. The energy-flux vector in strongly rotating turbulence is demonstrated based on the energy transfer rate obtained by direct numerical simulations. It is found that the direction of the energy-flux vector is consistent with the prediction of the weak turbulence theory in the wavenumber range dominated by inertial waves. However, the energy flux along the critical wavenumbers predicted by the critical balance in the buffer range between the weak turbulence range and the isotropic Kolmogorov turbulence range is not observed in the present simulations. This discrepancy between the critical balance and the present numerical results is discussed and dissipation is found to play an important role in the energy flux in the buffer range.

As far as plastron is sustained, superhydrophobic (SHPo) surfaces are expected to reduce skin-friction drag in any flow conditions including large-scale turbulent boundary-layer flows of marine vessels. However, despite many successful drag reductions reported using laboratory facilities, the plastron on SHPo surfaces was persistently lost in high-Reynolds-number flows on open water, and no reduction has been reported until a recent study using certain microtrench SHPo surfaces underneath a boat (Xu et al., Phys. Rev. Appl., vol. 13, no. 3, 2020, 034056). Since scientific studies with controlled flows are difficult with a boat on ocean water, in this paper we test similar SHPo surfaces in a high-speed towing tank, which provides well-controlled open-water flows, by developing a novel $0.7\ \textrm {m} \times 1.4\ \textrm {m}$ towing plate, which subjects a $4\ \textrm {cm} \times 7\ \textrm {cm}$ sample to the high-Reynolds-number flows of the plate. In addition to the 7 cm long microtrenches, trenches divided into two in length are also tested and reveal an improvement. The skin-friction drag ratio relative to a smooth surface is found to be decreasing with increasing Reynolds number, down to 73 % (i.e. 27 % drag reduction) at $Re_x\sim 8\times 10^6$, before starting to increase at higher speeds. For a given gas fraction, the trench width non-dimensionalized to the viscous length scale is found to govern the drag reduction, in agreement with previous numerical results.

The present study investigates the fundamental mechanisms of interaction between the propeller wake vortices and an untipped non-lifting wing. The study consists of a comprehensive experimental survey of a reference propeller–wing configuration with a high thickness parameter and is based on time-resolved visualisations and detailed flow and wall-pressure measurements. The experiment was designed to investigate the dynamics of the propeller blade vortices during the approach, encounter and penetration phases of the interaction and downstream of the body. To this end, three different models of the wing were manufactured including a transparent Perspex model that was crucial to simultaneously visualise the evolution of the vortex branches on the pressure and suction side of the body during the penetration phase. The study gains insight into the fundamental underlying mechanisms of the complex interaction between the propeller tip and blade trailing vortices and the wing for different propeller loadings. It is found that, during the encounter and the early penetration phases, tip vortex behaviour is strongly influenced by its interaction with the boundary layer of the wing that is manifested by a non-symmetrical evolution and breakdown of the vortex portions travelling along the pressure and suction sides of the wing. Reconnection between the vortex lines originating within the vortex core and the wing boundary layer maintains the linkage between the pressure and suction side portions of the vortex during the penetration phase and drives their rejoining downstream of the wing.

We derive a Noether current for the Eulerian variational principle of ideal non-barotropic magnetohydrodynamics (MHD). It was shown previously that ideal non-barotropic MHD is mathematically equivalent to a five function field theory with an induced geometrical structure in the case that field lines cover surfaces and this theory can be described using a variational principle. Here we use various symmetries of the flow to derive topological constants of motion through the derived Noether current and discuss their implication for non-barotropic MHD.

This paper presents a quantitative evaluation of the Reynolds analogy factor $s=2c_h/c_f$ for self-similar turbulent boundary layers with pressure gradients in the streamwise direction via direct numerical simulation. Both sub- and supersonic cases are considered at nearly adiabatic wall conditions. The Reynolds analogy factor is found to be greatly increased for adverse-pressure-gradient cases and decreased for favourable-pressure-gradient cases. Although the boundary layers considered cover a comparatively large Reynolds-number range from small to moderate Reynolds numbers, no Reynolds-number dependency of $s$ is found in the parameter range investigated. Mach-number influences on $s$ are found to be small; $s$ decreases slightly with increasing Mach number. The influence of pressure gradients on $s$ turns out to be well approximated by the analytical relation derived by So (Intl J. Heat Mass Transfer, vol. 37, 1994, pp. 27–41) for incompressible flow if a fixed, calibrated Reynolds number is used. Moreover, the effects of non-self-similarity prior to the self-similar region are assessed.

We study the dynamic wetting of a self-propelled viscous droplet using the time-dependent lubrication equation on a conical-shaped substrate for different cone radii, cone angles and slip lengths. The droplet velocity is found to increase with the cone angle and the slip length, but decrease with the cone radius. We show that a film is formed at the receding part of the droplet, much like the classical Landau–Levich–Derjaguin film. The film thickness $h_f$ is found to decrease with the slip length $\lambda$. By using the approach of matching asymptotic profiles in the film region and the quasi-static droplet, we obtain the same film thickness as the results from the lubrication approach for all slip lengths. We identify two scaling laws for the asymptotic regimes: $h_fh''_o \sim Ca^{2/3}$ for $\lambda \ll h_f$ and $h_f h''^{3}_o\sim (Ca/\lambda )^2$ for $\lambda \gg h_f$; here, $1/h''_o$ is a characteristic length at the receding contact line and $Ca$ is the capillary number. We compare the position and the shape of the droplet predicted from our continuum theory with molecular dynamics simulations, which are in close agreement. Our results show that manipulating the droplet size, the cone angle and the slip length provides different schemes for guiding droplet motion and coating the substrate with a film.

Advective trapping occurs when solute enters low velocity zones in heterogeneous porous media. Classical local modelling approaches combine the impact of slow advection and diffusion into a hydrodynamic dispersion coefficient and many temporally non-local approaches lump these mechanisms into a single memory function. This joint treatment makes parameterization difficult and thus prediction of large-scale transport a challenge. Here, we investigate the mechanisms of advective trapping and their impact on transport in media composed of a high conductivity background and isolated low permeability inclusions. Breakthrough curves show that effective transport changes from a streamtube-like behaviour to genuine random trapping as the degree of disorder of the inclusion arrangement increases. We upscale this behaviour using a Lagrangian view point, in which idealized solute particles transition over a fixed distance at random advection times combined with Poissonian advective trapping events. We discuss the mathematical formulation of the upscaled model in the continuous time random walk and mobile–immobile mass transfer frameworks, and derive a model for large-scale solute non-Fickian dispersion. These findings give new insight into transport in highly heterogeneous media.

We present simulations of a circular cylinder undergoing vortex-induced vibration in-line with a free stream in conjunction with a theory for the fluid dynamics. Initially, it is shown that increasing the Reynolds number from 100 to 250 results in a 12-fold increase of the peak response amplitude at a fixed mass ratio of $m^*=5$. Subsequently, $m^*$ is varied from 2 up to 20 at a fixed Reynolds number of 180. The response amplitude as a function of the reduced velocity $U^*$ displays a single excitation region with peak amplitudes of approximately 1% of the cylinder diameter, irrespective of the $m^*$ value. The vibration is always excited by the alternating shedding of single vortices. We develop a new model for the in-line fluid force, which comprises an inviscid inertial force, a quasi-steady drag and a wake drag induced by vortex shedding. Our analysis shows that the wake drag appropriately captures a gradual shift in the timing of vortex shedding in its phase variation as a function of $U^*$ while the magnitude of the wake drag displays a resonant amplification within the excitation region. We use the theory to illustrate why peak amplitudes, which occur when the vibration frequency is equal to the structural frequency in still fluid, do not depend on $m^*$, in agreement with our simulations as well as previous experiments at Reynolds numbers higher than considered here. This new theory provides physical insight which could not be attained heretofore by employing semi-empirical approaches in the literature.

In resolvent analyses of turbulent channel flows it has been common practice to neglect or model the nonlinear forcing term that forms the input of the resolvent. However, the spatiotemporal structure of this term is mostly unknown. Here, this nonlinear forcing term is quantified. The Fourier transform of its two-point space–time correlation, its cross-spectral density (CSD), is computed. The CSD is evaluated for two channel flows at friction Reynolds numbers $Re_{\tau } =179$ and $Re_{\tau } =543$ via direct numerical simulations (DNS). The CSDs are computed for energetic structures typical of buffer-layer and large-scale motions, for different temporal frequencies. It is found that the forcing is structured and that its solenoidal part, which is the only one affecting the velocity field, is the combination of an oblique streamwise vortical forcing and a streamwise component that counteract each other, as in a destructive interference. It is shown that a rank-2 approximation of the forcing, with only the most energetic spectral proper orthogonal decomposition (SPOD) modes, leads to the bulk of the response. Moreover, it is found that the nonlinear forcing term has a non-negligible projection onto the linear sub-optimal forcings of resolvent analysis, which demonstrates that the linear optimal forcing is not representative of the nonlinear forcing. Finally, it is clarified that the Cess eddy-viscosity-modelled forcing improves the accuracy of resolvent analysis prediction because the modelled forcing projects onto the linear sub-optimal forcings similarly to DNS data.

The dynamics of thin volatile droplets comprising of binary mixtures deposited on a heated substrate are investigated. Using lubrication theory, we develop a novel one-sided model to predict the spreading and retraction of an evaporating sessile axisymmetric droplet formed of a volatile binary mixture on a substrate with high wettability. A thin droplet with a moving contact line is considered, taking into account the variation of liquid properties with concentration as well as the effects of inertia. The parameter space is explored and the resultant effects on wetting and evaporation are evaluated. Increasing solutal Marangoni stress enhances spreading rates in all cases, approaching those of superspreading liquids. To validate our model, experiments are conducted with binary ethanol–water droplets spreading on hydrophilic glass slides heated from below. The spreading rate is quantified, revealing that preferential evaporation of the more volatile component (ethanol) at the contact line drives superspreading, leading in some cases to a contact line instability. Good qualitative agreement is found between our model and experiments, with quantitative agreement being achieved in terms of spreading rate.

We consider the Brownian motion of a rigid spherical particle in a binary fluid mixture, which lies in the homogeneous phase near the demixing critical point, assuming that neither component is more attracted by the particle surface. In a recent study, it was experimentally shown that the self-diffusion coefficient first decreases and then reaches a plateau as the critical point is approached. The decrease should reflect the critical enhancement of the viscosity, while the plateau was interpreted as representing the suppression of the enhancement due to the shear around the particle. To examine this interpretation, we take into account the local shear rate to calculate the dependence of the drag coefficient on the particle speed, and then utilize a Langevin equation to calculate the self-diffusion coefficient.

We examine the flow resulting from a vertically distributed wall-source plume in both an unventilated and ventilated space. First, we present experimental ambient buoyancy measurements for an unventilated ‘filling box’ where the developing ambient buoyancy profiles are successfully modelled using an adapted ‘peeling’ model which incorporates results presented in Part 1 of this work. We then present steady-state ambient buoyancy measurements for a ventilated box. Using dye visualisation, it is observed that, in the steady state, negligible ambient vertical transport occurs within the stratified region, implying a linear ambient buoyancy stratification within this region, and we predict the gradient of this linear stratification. Finally, we apply our experimental results to two practical examples. We present a methodology to create a given linear ambient temperature stratification within a room via a prescribed uniform wall heat flux and consider the resulting temperature stratification within a large ventilated atrium with a wall heated by solar radiation.

We examine a turbulent distributed wall-source plume: the flow resulting from a uniform vertical wall source of buoyancy such as that produced by an evenly heated or cooled vertical wall. The vertically distributed buoyancy source is created by forcing dense salt water solution through a porous wall. Velocity measurements on a vertical plane normal to the wall are first presented examining the full height of the wall in order to identify the region in which the bulk flow has become fully turbulent, self-similar and reached an invariant balance between the fluxes of volume, momentum and buoyancy. Simultaneous velocity and buoyancy field measurements are then presented in this region and an entrainment coefficient of $\alpha = 0.068 \pm 0.006$ is determined. This value is small compared to that of buoyancy-driven unbounded flows, e.g. a free line plume, and we reason this to be due to the presence of a rigid boundary restricting meandering and turbulence production, rather than the effect of the vertically distributed source of buoyancy. Turbulent velocity and buoyancy statistics are presented and, in order to gain physical insights into the flow behaviour, the results are compared to those of other canonical buoyancy-driven free and wall-bounded flows. We show that the bulk mixing of distributed wall-source plumes can be captured by consideration of the characteristic vertical velocities and a constant entrainment coefficient. This mixing is inhibited both by the presence of a rigid boundary and the reduced characteristic velocities (compared to those of wall line plumes).

This study examines the flow of dense granular materials under external shear stress and pressure using discrete element method simulations. In this method, the material is allowed to strain along all periodic directions and adapt its solid volume fraction in response to an imbalance between the internal state of stress and the external applied stress. By systematically varying the external shear stress and pressure, the steady rheological response is simulated for: (1) rate-independent quasi-static flow; and (2) rate-dependent inertial flow. The simulated flow is viscometric with non-negligible first and second normal stress differences. While both normal stress differences are negative in inertial flows, the first normal stress difference switches from negative to slightly positive, and second normal stress difference tends to zero in quasi-static flows. The first normal stress difference emerges from a lack of coaxiality between a second-rank contact fabric tensor and strain rate tensor in the flow plane, while the second normal stress difference is linked to an excess of contacts in the shear plane compared with the vorticity direction. A general rheological model of second order (in terms of strain rate tensor) is proposed to describe the two types of flow, and the model is calibrated for various values of interparticle friction from simulations on nearly monodisperse spheres. The model incorporates normal stress differences in both regimes of flow and provides a complete viscometric description of steady dense granular flows.

We use well resolved numerical simulations with the lattice Boltzmann method to study Rayleigh–Bénard convection in cells with a fractal boundary in two dimensions for $Pr = 1$ and $Ra \in \left [10^7, 10^{10}\right ]$, where Pr and Ra are the Prandtl and Rayleigh numbers. The fractal boundaries are functions characterized by power spectral densities $S(k)$ that decay with wavenumber, $k$, as $S(k) \sim k^{p}$ ($p < 0$). The degree of roughness is quantified by the exponent $p$ with $p < -3$ for smooth (differentiable) surfaces and $-3 \le p < -1$ for rough surfaces with Hausdorff dimension $D_f=\frac {1}{2}(p+5)$. By computing the exponent $\beta$ using power law fits of $Nu \sim Ra^{\beta }$, where $Nu$ is the Nusselt number, we find that the heat transport scaling increases with roughness through the top two decades of $Ra \in \left [10^8, 10^{10}\right ]$. For $p$$= -3.0$, $-2.0$ and $-1.5$ we find $\beta = 0.288 \pm 0.005, 0.329 \pm 0.006$ and $0.352 \pm 0.011$, respectively. We also find that the Reynolds number, $Re$, scales as $Re \sim Ra^{\xi }$, where $\xi \approx 0.57$ over $Ra \in \left [10^7, 10^{10}\right ]$, for all $p$ used in the study. For a given value of $p$, the averaged $Nu$ and $Re$ are insensitive to the specific realization of the roughness.

Cross-stream migration of a droplet in an incipient flow turns out to be of outstanding importance in several emerging applications encompassing chemistry, engineering and biology. Here, we bring out the confluence of confinement, oscillatory axial pressure gradient and steady axial electric field towards controlling spatiotemporal characteristics of cross-stream migration of droplets in a micro-confined fluidic environment, bearing immense implications in in vitro modelling of bio-analytical procedures. Under the sole influence of an oscillatory axial pressure gradient, the time taken by a droplet to achieve a steady-state transverse position is significantly long and the direction of the droplet's motion cannot be altered at will. However, confinement-modulated electrohydrodynamic interactions enable overcoming this constraint, even when the applied electric field is orthogonal to the intended direction of droplet migration, a proposition that is not feasible in an unbounded domain. Our results reveal that depending on the relative electrical properties of the droplet and the carrier phases and a competing influence of electrical, viscous and capillary stresses, the rate of transverse migration can be controlled by effectively modulating the axial oscillations in its cross-stream motion. Beyond a threshold value of the applied electric field, simultaneous enhancement in the droplet migration rate and reversal in the direction of its lateral migration become possible, which cannot otherwise be achieved by the oscillatory pressure field alone. Furthermore, the oscillatory characteristics in the droplet migration can be dampened out completely by exploiting the addressed physical interplay. Results from in-house experiments corroborate our theoretical conjecture.

A fully nonlinear Cauchy–Poisson problem is investigated analytically by a small-time expansion. The inviscid incompressible fluid layer has an initially horizontal surface. The fluid is forced into motion by an impulsive surface pressure. The early nonlinear free-surface problem is solved to second order in a small-time expansion by the Lagrangian description of motion. Comparisons are made with two other solution procedures for the same nonlinear problem in the absence of gravity: a third-order small-time expansion and a numerical solution, based on full nonlinearity according to the standard Eulerian description. Good agreement is found between the present second-order Lagrangian solution and the previous third-order Eulerian solution, until both these asymptotic expansions diverge rather abruptly at the same time.

We investigate the large-scale circulation (LSC) of turbulent Rayleigh–Bénard convection in a large box of aspect ratio $\varGamma =32$ for Rayleigh numbers up to $Ra=10^9$ and at a fixed Prandtl number $Pr=1$. A conditional averaging technique allows us to extract statistics of the LSC even though the number and the orientation of the structures vary throughout the domain. We find that various properties of the LSC obtained here, such as the wall-shear stress distribution, the boundary layer thicknesses and the wind Reynolds number, do not differ significantly from results in confined domains ($\varGamma \approx 1$). This is remarkable given that the size of the structures (as measured by the width of a single convection roll) more than doubles at the highest $Ra$ as the confinement is removed. An extrapolation towards the critical shear Reynolds number of $Re_s^{{crit}} \approx 420$, at which the boundary layer (BL) typically becomes turbulent, predicts that the transition to the ultimate regime is expected at $Ra_{{crit}} \approx {O}(10^{15})$ in unconfined geometries. This result is in line with the Göttingen experimental observations (He et al., Phys. Rev. Lett., vol. 108, 2012, 024502; New J. Phys., vol. 17, 2015, 063028). Furthermore, we confirm that the local heat transport close to the wall is highest in the plume impacting region, where the thermal BL is thinnest, and lowest in the plume emitting region, where the thermal BL is thickest. This trend, however, weakens with increasing $Ra$.

A nonlinear Cauchy–Poisson problem with impulsive surface forcing is investigated analytically and numerically. An incompressible liquid with an initially horizontal surface is instantaneously put into motion by an impulsive surface pressure distribution turned on and off during an infinitesimal time interval. We consider symmetric, antisymmetric and asymmetric pressure impulses based on dipoles and quadrupoles. The subsequent inviscid free-surface flow is governed by fully nonlinear surface conditions, which are solved exactly to third order in a small-time expansion. The small-time expansion applies to flows dominated by inertia. Such flows are generated by relatively strong pressure impulses, measured in gravitational units. We solve the problem numerically and find that only relatively weak pressure impulses will lead to oscillatory waves. The free surface will break before a full gravitational oscillation is completed when the amplitude of the pressure impulse exceeds one gravitational unit.

A slowly varying modes solution of Wentzel–Kramers–Brillouin type is derived for the problem of sound propagation in a slowly varying two-dimensional duct with homentropic inviscid sheared mean flow and acoustically lined walls of slowly varying impedance. The modal shape function and axial wavenumber are described by the Pridmore-Brown eigenvalue equation. The slowly varying modal amplitude is determined in the usual way by an equation resulting from a solvability condition. For a general mean flow, this equation can be solved in the form of an incomplete adiabatic invariant. Due to conservation of specific mean vorticity along streamlines, two simplifications prove possible for a linearly sheared mean flow: (i) an analytically exact approximation for the mean flow, and (ii) a complete adiabatic invariant for the acoustics. For this last configuration some example cases are evaluated numerically, where the Pridmore-Brown eigenvalue problem is solved by a Galerkin projection combined with an efficient nonlinear iteration.