The atmospheric boundary layer is the level of the atmosphere where all human activities occur. It is a layer characterized by its turbulent flow state, meaning that the velocity, temperature and scalar concentrations fluctuate over scales that range from less than a millimetre to several kilometres. It is those fluctuations that make dispersion of pollutants and transport of heat, momentum as well as scalars such as carbon dioxide or cloud-condensation nuclei efficient. It is also the layer where a ‘hand-shake’ occurs between activities on the land surface and the climate system, primarily due to the action of large energetic swirling motions or eddies. The atmospheric boundary layer experiences dramatic transitions depending on whether the underlying surface is being heated or cooled. The existing paradigm describing the size and energetics of large-scale and very large-scale eddies in turbulent flows has been shaped by decades of experiments and simulations on smooth pipes and channels with no surface heating or cooling. The emerging picture, initiated by A. A. Townsend in 1951, is that large- and very large-scale motions appear to be approximated by a collection of hairpin-shaped vortices whose population density scales inversely with distance from the boundary. How does surface heating, quintessential to the atmospheric boundary layer, alter this canonical picture? What are the implications of such a buoyancy force on the geometry and energy distribution across velocity components in those large eddies? How do these large eddies modulate small eddies near the ground? Answering these questions and tracking their consequences to existing theories used today to describe the flow statistics in the atmospheric boundary layer are addressed in the work of Salesky & Anderson (J. Fluid Mech., vol. 856, 2018, pp. 135–168). The findings are both provocative and surprisingly simple.

]]>We develop a physical and computational model for performing fully coupled, grain-resolved direct numerical simulations of cohesive sediment, based on the immersed boundary method. The model distributes the cohesive forces over a thin shell surrounding each particle, thereby allowing for the spatial and temporal resolution of the cohesive forces during particle–particle interactions. The influence of the cohesive forces is captured by a single dimensionless parameter in the form of a cohesion number, which represents the ratio of cohesive and gravitational forces acting on a particle. We test and validate the cohesive force model for binary particle interactions in the drafting–kissing–tumbling (DKT) configuration. Cohesive sediment grains can remain attached to each other during the tumbling phase following the initial collision, thereby giving rise to the formation of flocs. The DKT simulations demonstrate that cohesive particle pairs settle in a preferred orientation, with particles of very different sizes preferentially aligning themselves in the vertical direction, so that the smaller particle is drafted in the wake of the larger one. This preferred orientation of cohesive particle pairs is found to remain influential for systems of higher complexity. To this end, we perform large simulations of 1261 polydisperse settling particles starting from rest. These simulations reproduce several earlier experimental observations by other authors, such as the accelerated settling of sand and silt particles due to particle bonding, the stratification of cohesive sediment deposits, and the consolidation process of the deposit. They identify three characteristic phases of the polydisperse settling process, viz. (i) initial stir-up phase with limited flocculation, (ii) enhanced settling phase characterized by increased flocculation, and (iii) consolidation phase. The simulations demonstrate that cohesive forces accelerate the overall settling process primarily because smaller grains attach to larger ones and settle in their wakes. For the present cohesive number values, we observe that settling can be accelerated by up to 29 %. We propose physically based parametrization of classical hindered settling functions introduced by earlier authors, in order to account for cohesive forces. An investigation of the energy budget shows that, even though the work of the collision forces is much smaller than that of the hydrodynamic drag forces, it can substantially modify the relevant energy conversion processes.

]]>Manifestly Markovian closures for the interaction of two-dimensional inhomogeneous turbulent flows with Rossby waves and topography are formulated and compared with large ensembles of direct numerical simulations (DNS) on a generalized -plane. Three versions of the Markovian inhomogeneous closure (MIC) are established from the quasi-diagonal direct interaction approximation (QDIA) theory by modifying the response function to a Markovian form and employing respectively the current-time (quasi-stationary) fluctuation dissipation theorem (FDT), the prior-time (non-stationary) FDT and the correlation FDT. Markov equations for the triad relaxation functions are derived that carry similar information to the time-history integrals of the non-Markovian QDIA closure but become relatively more efficient for long integrations. Far from equilibrium processes are studied, where the impact of a westerly mean flow on a conical mountain generates large-amplitude Rossby waves in a turbulent environment, over a period of 10 days. Excellent agreement between the evolved mean streamfunction and mean and transient kinetic energy spectra are found for the three versions of the MIC and two variants of the non-Markovian QDIA compared with an ensemble of 1800 DNS. In all cases mean Rossby wavetrain pattern correlations between the closures and the DNS ensemble are greater than 0.9998.

]]>Damped internal wave beams in stratified fluids have long been known to generate strong mean flows through a mechanism analogous to acoustic streaming. While the role of viscous boundary layers in acoustic streaming has been thoroughly addressed, it remains largely unexplored in the case of internal waves. Here we compute the mean flow generated close to an undulating wall that emits internal waves in a viscous, linearly stratified two-dimensional Boussinesq fluid. Using a quasi-linear approach, we demonstrate that the form of the boundary conditions dramatically impacts the generated boundary streaming. In the no-slip scenario, the early-time Reynolds stress divergence within the viscous boundary layer is much stronger than within the bulk while also driving flow in the opposite direction. Whatever the boundary condition, boundary streaming is however dominated by bulk streaming at larger time. Using a Wentzel–Kramers–Brillouin approach, we investigate the consequences of adding boundary streaming effects to an idealised model of wave–mean flow interactions known to reproduce the salient features of the quasi-biennial oscillation. The presence of wave boundary layers has a quantitative impact on the flow reversals.

]]>We study numerically the dynamics of an insoluble surfactant-laden droplet in a simple shear flow taking surface viscosity into account. The rheology of drop surface is modelled via a Boussinesq–Scriven constitutive law with both surface tension and surface viscosity depending strongly on the surface concentration of the surfactant. Our results show that the surface viscosity exhibits non-trivial effects on the surfactant transport on the deforming drop surface. Specifically, both dilatational and shear surface viscosity tend to eliminate the non-uniformity of surfactant concentration over the drop surface. However, their underlying mechanisms are entirely different; that is, the shear surface viscosity inhibits local convection due to its suppression on drop surface motion, while the dilatational surface viscosity inhibits local dilution due to its suppression on local surface dilatation. By comparing with previous studies of droplets with surface viscosity but with no surfactant transport, we find that the coupling between surface viscosity and surfactant transport induces non-negligible deviations in the dynamics of the whole droplet. More particularly, we demonstrate that the dependence of surface viscosity on local surfactant concentration has remarkable influences on the drop deformation. Besides, we analyse the full three-dimensional shape of surfactant-laden droplets in simple shear flow and observe that the drop shape can be approximated as an ellipsoid. More importantly, this ellipsoidal shape can be described by a standard ellipsoidal equation with only one unknown owing to the finding of an unexpected relationship among the drop’s three principal axes. Moreover, this relationship remains the same for both clean and surfactant-laden droplets with or without surface viscosity.

]]>In this investigation, a data-driven turbulence closure framework is introduced and deployed for the subgrid modelling of Kraichnan turbulence. The novelty of the proposed method lies in the fact that snapshots from high-fidelity numerical data are used to inform artificial neural networks for predicting the turbulence source term through localized grid-resolved information. In particular, our proposed methodology successfully establishes a map between inputs given by stencils of the vorticity and the streamfunction along with information from two well-known eddy-viscosity kernels. Through this we predict the subgrid vorticity forcing in a temporally and spatially dynamic fashion. Our study is both a priori and a posteriori in nature. In the former, we present an extensive hyper-parameter optimization analysis in addition to learning quantification through probability-density-function-based validation of subgrid predictions. In the latter, we analyse the performance of our framework for flow evolution in a classical decaying two-dimensional turbulence test case in the presence of errors related to temporal and spatial discretization. Statistical assessments in the form of angle-averaged kinetic energy spectra demonstrate the promise of the proposed methodology for subgrid quantity inference. In addition, it is also observed that some measure of a posteriori error must be considered during optimal model selection for greater accuracy. The results in this article thus represent a promising development in the formalization of a framework for generation of heuristic-free turbulence closures from data.

]]>Direct numerical simulations are used to characterize wind-shear effects on entrainment in a barotropic convective boundary layer (CBL) that grows into a linearly stratified atmosphere. We consider weakly to strongly unstable conditions , where is the encroachment CBL depth and is the Obukhov length. Dimensional analysis allows us to characterize such a sheared CBL by a normalized CBL depth, a Froude number and a Reynolds number. The first two non-dimensional quantities embed the dependence of the system on time, on the surface buoyancy flux, and on the buoyancy stratification and wind velocity in the free atmosphere. We show that the dependence of entrainment-zone properties on these two non-dimensional quantities can be expressed in terms of just one independent variable, the ratio between a shear scale and a convective scale , where is the velocity increment across the entrainment zone, and is the buoyancy frequency of the free atmosphere. Here and represent the entrainment-zone thickness in the limits of weak convective instability (strong wind) and strong convective instability (weak wind), respectively. We derive scaling laws for the CBL depth, the entrainment-zone thickness, the mean entrainment velocity and the entrainment-flux ratio as functions of . These scaling laws can also be expressed as functions of only a Richardson number , but not in terms of only the stability parameter .

]]>We study the flow and transport of heat or mass, modelled as passive scalars, within a basic geometrical unit of a three-dimensional multipolar flow – a triangular prism – characterised by a side length , a normalised thickness and an apex angle , and connected to inlet and outlet pipes of equal normalised radius perpendicularly to the plane of the flow. The flow and scalar fields are investigated over the range and , where and are respectively the Reynolds and Péclet numbers imposed at the inlet pipe when either a Dirichlet ( ) or a Neumann ( ) scalar boundary condition is imposed at the wall unattached to the inlets and outlets. A scalar no-flux boundary condition is imposed at all the other walls. An axisymmetric model is applied to understand the flow and scalar transport in the inlet and outlet regions, which consist of a turning region close to the pipe centreline and a channel region away from it. A separate two-dimensional model is then developed for the channel region by solving the integral form of the momentum and scalar advection–diffusion equations. Analytical relations between geometrical, flow and scalar transport parameters based on similarity and integral methods are generated and agree closely with numerical solutions. Finally, three-dimensional numerical calculations are undertaken to test the validity of the axisymmetric and depth-averaged analyses. Dominant flow and scalar transport features vary dramatically across the flow domain. In the turning region, the flow is a largely irrotational straining flow when and a dominantly viscous straining flow when . The thickness of the scalar boundary layer scales to the local Péclet number to the power . The diffusive flux and the scalar at the wall where ( ) or ( ) is imposed, respectively, are constant. In the channel region, the flow is parabolic and dominated by a source flow near the inlet and an irrotational straining flow away from it. When is imposed the scalar decreases exponentially with distance from the inlet and the normalised scalar transfer coefficient converges to . When is imposed, varies proportionally to surface area. Transport in the straining region downstream of the inlet is diffusion-limited, and and are functions of the geometrical parameters , , and . In addition to describing the fundamental properties of the flow and passive transport in multipolar configurations, the present work demonstrates how geometrical and flow parameters should be set to control transfers in the different regions of the flow domain.

]]>Large-scale coherent structures such as jets in Rayleigh–Bénard convection and related systems are receiving increasing attention. This paper studies, both numerically and theoretically, the process of jet formation in two-dimensional salt-finger convection. The approach utilizes an asymptotically derived system of equations referred to as the modified Rayleigh–Bénard convection (MRBC) model, valid in the geophysically and astrophysically relevant limit in which the solute diffuses much more slowly than heat. In these equations, convection is driven by a destabilizing salinity gradient while the effects of the stabilizing temperature gradient manifest themselves as an additional anisotropic dissipation acting on large scales. The MRBC system is specified by two external parameters: the Schmidt number (ratio of viscosity to solutal diffusivity) and the Rayleigh ratio (ratio between the Rayleigh numbers of the destabilizing solutal stratification and the stabilizing thermal stratification). Two distinct regimes are explored for fixed . In all cases studied the system develops a horizontal jet structure that is maintained self-consistently by turbulent fluctuations, but coarsens over time. For intermediate Rayleigh ratios (e.g. ), the MRBC model captures the relaxation oscillations superposed on the jet structure observed at similar parameter values in direct numerical simulations of the primitive equations. For smaller Rayleigh ratios (e.g. ), a regime for which direct numerical simulation of the primitive equations is difficult because of the presence of fast gravity waves, the MRBC model reveals the existence of statistically steady jets whose properties are studied in detail. Three hierarchical models, the MRBC and further reductions in the form of quasilinear and single-mode approximations, are used to confirm that jets form and are sustained as a result of the interaction between fluctuations (salt fingers) and large-scale horizontally averaged horizontal flows (jets). Even though the small-scale structures exhibited by the three models exhibit clear differences, all three produce the same power-law spectrum of the mean fields at large vertical scales: in all, the spectrum of the mean streamfunction scales as and the mean salinity field scales as , with the vertical wavenumber. A theoretical explanation of these observations based on the dominant balances in the mean and fluctuation equations is provided. As a consequence, the jets have a zigzag profile, a conclusion that is consistent with numerical simulations. Based on numerical observations, a three-component phenomenological model consisting of a linearly growing mode, a linearly damped mode and a mean mode is proposed to explain the observed transition from statistically steady jet structure to jets with superposed oscillations that takes place with increasing Rayleigh ratio.

]]>A direct numerical simulation study is conducted to investigate sinusoidal oscillatory flow over a two-dimensional wavy wall. The height and wavelength of the bottom profile, and the period and amplitude of the free-stream oscillation, are selected to mimic a wave-driven boundary layer over vortex ripples on a sandy seabed. Two cases with different Reynolds numbers are considered, and the higher- case achieves a fully developed turbulent state with a wide separation between the energy-containing and dissipative scales. The oscillatory flow is characterized by coherent columnar vortices, which are the main transport agents of turbulent kinetic energy and enstrophy. Two classes of coherent vortices are observed: (i) a primary vortex formed at the lee side of the ripple by flow separation at the crest; (ii) a secondary vortex formed beneath the primary vortex by vortex-induced separation. When the free-stream velocity weakens, these vortices form a counter-rotating vortex dipole and eject themselves over the crest with their mutual induction. Turbulence production peaks twice in a half-cycle; during the formation of the primary vortex and during the ejection of the vortex dipole. The intensity of the former peak remains low in the lower- case, as the vortex dipole follows a higher altitude trajectory limiting its interactions with the bottom, and leaving minimal residual turbulence around the ripples for the subsequent half-cycle. Flow snapshots and spectral analysis reveal two dominant three-dimensional features: (i) an energetic vortex mode with a preferred spanwise wavelength close to the ripple wavelength; (ii) streamwise vortical structures in near-wall regions with a relatively shorter spanwise spacing influenced by viscous effects. The vortex mode becomes strong when the cores of the vortices are strained to an elliptical form while moving towards the crest. Following the detachment of the vortices from the ripple, the vortex mode in the higher- case breaks down the spanwise coherence of the columnar vortices and decomposes them into intermittent patches of turbulent vortex clusters. The distribution of wall shear stress over the ripple is also analysed in detail. The peak values are observed near the ripple crest around the ejection of the vortex dipole and the maximum free-stream velocity. In the former, both the vortex mode and streamwise vortices have strong footprints on the wall, yielding a bimodal wall-shear-stress spectrum with two distinctive peaks. In the second high-stress regime, decaying coherent vortices impose strong inhomogeneity on the wall shear stress as their wall-attached parts sweep the ripples. These spanwise variations in the wall shear provide insights into the instability of two-dimensional sand ripples.

]]>In this paper, flow over a streamwise oscillating circular cylinder is numerically simulated to examine the effects of the driving amplitude and frequency on the distribution of the lock-in regions in laminar flows. At , lock-in is categorized according to the spectral features of the lift coefficient as two different lock-in phenomena: harmonic and subharmonic lock-in. These lock-in phenomena are represented as maps on the driving amplitude–frequency plane, which have subharmonic lock-in regions and two harmonic lock-in regions. The frequency range of the subharmonic region is shifted to lower frequencies with increasing amplitude, and the lower boundary of this subharmonic region is successfully predicted. A symmetric harmonic region with a symmetric vortex pattern is observed in a certain velocity range for a moving cylinder. Aerodynamic features induced by different flow patterns in each region are presented on the driving amplitude–frequency plane. The lock-in region and aerodynamic features at and are compared with the results for . A subharmonic region and two harmonic regions are observed at , and these show the same features as for at a low driving amplitude. Lock-in at also shows one subharmonic region and two harmonic regions. However, compared with the case, the symmetric harmonic lock-in is dominant. The features of aerodynamic force at and are represented on a force map, which shows similar characteristics in corresponding regions for the case.

]]>We propose an alternative to the prevailing framework for modelling tear-film breakup, which posits a layered structure with a mucus layer next to the cornea and an aqueous layer on top. Experimental evidence shows continuous variation of mucin concentration throughout the tear film, with no distinct boundary between the two layers. Thus, we consider a continuous-viscosity model that replaces the mucus and aqueous layers by a single liquid layer with continuous profiles of mucin concentration and viscosity, which are governed by advection–diffusion of mucin. The lipids coating the tear film are treated as insoluble surfactants as previously, and slip is allowed on the ocular surface. Using the thin-film approximation, we carry out linear stability analysis and nonlinear numerical simulations of tear-film breakup driven by van der Waals attraction. Results show that for the same average viscosity, having more viscous material near the ocular surface stabilizes the film and prolongs the breakup time. Compared with the layered models, the continuous-viscosity model predicts film breakup times that are in better agreement with experimental data. Finally, we also suggest a hydrodynamic explanation for how pathological loss of membrane-associated mucins may lead to faster breakup.

]]>We consider the problem of formulating perturbative expansions of the conformation tensor, which is a positive definite tensor representing polymer deformation in viscoelastic flows. The classical approach does not explicitly take into account that the perturbed tensor must remain positive definite – a fact that has important physical implications, e.g. extensions and compressions are represented similarly to within a negative sign, when physically the former are unbounded and the latter are bounded from below. Mathematically, the classical approach assumes that the underlying geometry is Euclidean, and this assumption is not satisfied by the manifold of positive definite tensors. We provide an alternative formulation that retains the conveniences of classical perturbation methods used for generating linear and weakly nonlinear expansions, but also provides a clear physical interpretation and a mathematical basis for analysis. The approach is based on treating a perturbation as a sequence of successively smaller deformations of the polymer. Each deformation is modelled explicitly using geodesics on the manifold of positive definite tensors. Using geodesics, and associated geodesic distances, is the natural way to model perturbations to positive definite tensors because it is consistent with the manifold geometry. Approximations of the geodesics can then be used to reduce the total deformation to a series expansion in the small perturbation limit. We illustrate our approach using direct numerical simulations of the nonlinear evolution of Tollmien–Schlichting waves.

]]>A macroscopic boundary condition to be used when a fluid flows over a rough surface is derived. It provides the slip velocity on an equivalent (smooth) surface in the form , where the dimensionless parameter is a measure of the roughness amplitude, denotes the strain-rate tensor associated with the outer flow in the vicinity of the surface and is a third-order slip tensor arising from the microscopic geometry characterizing the rough surface. This boundary condition represents the tensorial generalization of the classical Navier slip condition. We derive this condition, in the limit of small microscopic Reynolds numbers, using a multi-scale technique that yields a closed system of equations, the solution of which allows the slip tensor to be univocally calculated, once the roughness geometry is specified. We validate this generalized slip condition by considering the flow about a rough sphere, the surface of which is covered with a hexagonal lattice of cylindrical protrusions. Comparisons with direct numerical simulations performed in both laminar and turbulent regimes allow us to assess the validity and limitations of this condition and of the mathematical model underlying the determination of the slip tensor .

]]>We study the evolution of a melting front between the solid and liquid phases of a pure incompressible material where fluid motions are driven by unstable temperature gradients. In a plane-layer geometry, this can be seen as classical Rayleigh–Bénard convection where the upper solid boundary is allowed to melt due to the heat flux brought by the fluid underneath. This free-boundary problem is studied numerically in two dimensions using a phase-field approach, classically used to study the melting and solidification of alloys, which we dynamically couple with the Navier–Stokes equations in the Boussinesq approximation. The advantage of this approach is that it requires only moderate modifications of classical numerical methods. We focus on the case where the solid is initially nearly isothermal, so that the evolution of the topography is related to the inhomogeneous heat flux from thermal convection, and does not depend on the conduction problem in the solid. From a very thin stable layer of fluid, convection cells appear as the depth – and therefore the effective Rayleigh number – of the layer increases. The continuous melting of the solid leads to dynamical transitions between different convection cell sizes and topography amplitudes. The Nusselt number can be larger than its value for a planar upper boundary, due to the feedback of the topography on the flow, which can stabilize large-scale laminar convection cells.

]]>The interaction of stationary streaks undergoing non-modal growth with modally unstable instability waves in a high Mach number boundary-layer flow is studied using numerical computations. The geometry and flow conditions are selected to match a relevant trajectory location from the ascent phase of the HIFiRE-1 flight experiment; namely, a half-angle, circular cone with mm nose radius, free-stream Mach number equal to , unit Reynolds number equal to and wall-to-adiabatic temperature ratio of approximately over most of the vehicle. This paper investigates the nonlinear evolution of initially linear optimal disturbances that evolve into finite-amplitude streaks, followed by an analysis of the modal instability characteristics of the perturbed, streaky boundary-layer flow. The investigation is performed with a stationary, full Navier–Stokes equations solver and the plane-marching parabolized stability equations (PSE), in conjunction with partial-differential-equation-based planar eigenvalue analysis. The overall effect of streaks is to reduce the peak amplification factors of instability waves, indicating a possible downstream shift in the onset of laminar–turbulent transition. The present study confirms previous findings that the mean-flow distortion of the nonlinear streak perturbation reduces the amplification rates of the Mack-mode instability. More importantly, however, the present results demonstrate that the spanwise varying component of the streak can produce a larger effect on the Mack-mode amplification. The analysis of planar and oblique Mack-mode waves modulated by the presence of the streaks shows that the planar Mack mode still dominates the instability characteristics of the flow. The study with selected azimuthal wavenumbers for the stationary streaks reveals that a wavenumber of approximately times larger than the optimal wavenumber is more effective in stabilizing the planar Mack-mode instabilities. In the absence of unstable first-mode waves for the present cold-wall condition, transition onset is expected to be delayed until the peak streak amplitude increases to nearly 35 % of the free-stream velocity, when intrinsic instabilities of the boundary-layer streaks begin to dominate the transition process. For streak amplitudes below that limit a significant net stabilization is achieved, yielding a potential transition delay that can exceed 100 % of the length of the laminar region in the uncontrolled case.

]]>High resolution large eddy simulations (LES) are performed to study the interaction of a stationary shock with fully developed turbulent flow. Turbulent statistics downstream of the interaction are provided for a range of weakly compressible upstream turbulent Mach numbers , shock Mach numbers and Taylor-based Reynolds numbers . The LES displays minimal Reynolds number effects once an inertial range has developed for . The inertial range scales of the turbulence are shown to quickly return to isotropy, and downstream of sufficiently strong shocks this process generates a net transfer of energy from transverse into streamwise velocity fluctuations. The streamwise shock displacements are shown to approximately follow a decay with wavenumber as predicted by linear analysis. In conjunction with other statistics this suggests that the instantaneous interaction of the shock with the upstream turbulence proceeds in an approximately linear manner, but nonlinear effects immediately downstream of the shock significantly modify the flow even at the lowest considered turbulent Mach numbers.

]]>The interaction of an acoustic wave with a stratified fluid can drive strong streaming flows owing to the baroclinic production of fluctuating vorticity, as recently demonstrated by Chini et al. (J. Fluid Mech., 744, 2014, pp. 329–351). In the present investigation, a set of wave/mean-flow interaction equations is derived that governs the coupled dynamics of a standing acoustic-wave mode of characteristic (small) amplitude and the streaming flow it drives in a thin channel with walls maintained at differing temperatures. Unlike classical Rayleigh streaming, the resulting mean flow arises at rather than at . Consequently, fully two-way coupling between the waves and the mean flow is possible: the streaming is sufficiently strong to induce rearrangements of the imposed background temperature and density fields, which modifies the spatial structure and frequency of the acoustic mode on the streaming time scale. A novel Wentzel–Kramers–Brillouin–Jeffreys analysis is developed to average over the fast wave dynamics, enabling the coupled system to be integrated strictly on the slow time scale of the streaming flow. Analytical solutions of the reduced system are derived for weak wave forcing and are shown to reproduce results from prior direct numerical simulations (DNS) of the compressible Navier–Stokes and heat equations with remarkable accuracy. Moreover, numerical simulations of the reduced system are performed in the regime of strong wave/mean-flow coupling for a fraction of the computational cost of the corresponding DNS. These simulations shed light on the potential for baroclinic acoustic streaming to be used as an effective means to enhance heat transfer.

]]>Three-dimensional shock wave reflection comprises flow physics that is significantly different from the well-documented two-dimensional cases in a number of aspects. The most important differentiating factor is the sweep of the shock system. In particular, this work examines the nature of flow fields in which there is a transition of shock reflection configuration in three-dimensional space. The flow fields investigated have been made to exist in the absence of edge effects influencing the shock interaction and transition, as found previously to exist in conventional double-wedge studies. In general, the shock configurations are those with central regular and peripheral Mach reflection portions. It is shown that the sweep angle of the portions on either side of the transition point is subject to a cusp, as per an analytical model that is developed. This is confirmed with the use of numerical models with additional evidence provided by experimental results using oblique shadow photography. Further application of the principles of three-dimensional shock analysis and those pertaining to the sweep cusp model yield important insights regarding the overall shock geometry and that at transition.

]]>The Arctic halocline is generally stable to the development of double-diffusive and dynamic instabilities – the two major sources of small-scale mixing in the mid-latitude oceans. Despite this, observations show the abundance of double-diffusive staircases in the Arctic Ocean, which suggests the presence of some destabilizing process facilitating the transition from smooth-gradient to layered stratification. Recent studies have shown that an instability can develop in such circumstances if weak static shear is present even when the flow is dynamically and diffusively stable. However, the impact of oscillating shear, associated with the presence of internal gravity waves, has not yet been addressed for the diffusive case. Through two-dimensional simulations of diffusive convection, we have investigated the impact of the magnitude and frequency of externally forced oscillatory shear on the thermohaline-shear instability. Simulations with stochastic shear – characterized by a continuous spectrum of frequencies from inertial to buoyancy – indicate that thermohaline layering does occur due to the presence of destabilizing modes (oscillations of near the buoyancy frequency). These simulations show that such layers appear as well-defined steps in the temperature and salinity profiles. Thus, the thermohaline-shear instability is a plausible mechanism for staircase formation in the Arctic and merits substantial future study.

]]>Recent studies reveal that at large friction Reynolds number the inertially dominated region of the turbulent boundary layer is composed of large-scale zones of nearly uniform momentum segregated by narrow fissures of concentrated vorticity. Experiments show that, when scaled by the boundary-layer thickness, the fissure thickness is , while the dimensional jump in streamwise velocity across each fissure scales in proportion to the friction velocity . A simple model that exploits these essential elements of the turbulent boundary-layer structure at large is developed. First, a master wall-normal profile of streamwise velocity is constructed by placing a discrete number of fissures across the boundary layer. The number of fissures and their wall-normal locations follow scalings informed by analysis of the mean momentum equation. The fissures are then randomly displaced in the wall-normal direction, exchanging momentum as they move, to create an instantaneous velocity profile. This process is repeated to generate ensembles of streamwise velocity profiles from which statistical moments are computed. The modelled statistical profiles are shown to agree remarkably well with those acquired from direct numerical simulations of turbulent channel flow at large . In particular, the model robustly reproduces the empirically observed sub-Gaussian behaviour for the skewness and kurtosis profiles over a large range of input parameters.

]]>The time-averaged flow dynamics of a suspended cylindrical canopy patch with a bulk diameter of is investigated using large-eddy simulations (LES). The patch consists of constituent solid circular cylinders of height and diameter , mimicking patchy vegetation suspended in deep water ( , where is the total flow depth). After validation against published data, LES of a uniform incident flow impinging on the canopy patch was conducted to study the effects of canopy density ( , by varying ) and bulk aspect ratio ( , by varying ) on the near-wake structure and adjustment of flow pathways. The relationships between patch geometry, local flow bleeding (three-dimensional redistribution of flow entering the patch) and global flow diversion (streamwise redistribution of upstream undisturbed flow) are identified. An increase in either or decreases/increases/increases bleeding velocities through the patch surface area along the streamwise/lateral/vertical directions, respectively. However, a volumetric flux budget shows that a larger causes a smaller proportion of the flow rate entering the patch to bleed out vertically. The global flow diversion is found to be determined by both the patch geometrical dimensions and the local bleeding which modifies the sizes of the patch-scale near wake. While loss of flow penetrating the patch increases monotonically with increasing , its partition into flow diversion around and beneath the patch shows a non-monotonic dependence. The spatial extents of the wake, the flow-diversion dynamics and the bulk drag coefficients of the patch jointly reveal the fundamental differences of flow responses between suspended porous patches and their solid counterparts.

]]>The long-wave, reduced-gravity, shallow-water equations (the semi-geostrophic equations) are used to study the outflow of a river into the ocean. While previous models have studied dynamics driven by gradients in density, the focus here is on the effects of potential vorticity anomaly (PVa). The river water is taken to have the same density as a finite-depth upper layer of oceanic fluid, but the two fluids have different, uniform, potential vorticities. Under these assumptions, the governing equations reduce to two first-order, nonlinear partial differential equations which are integrated numerically for a prescribed efflux of river water and PVa. Results are found to depend strongly on the sign of the PVa, with all fluid turning downstream (in the direction of Kelvin-wave propagation) when the river water has positive PVa and anticyclonic flow upstream of the river mouth when the PVa is negative. In all cases, a nonlinear Kelvin wave propagates at finite speed ahead of the river water. Away from the river mouth, the uniformity of one set of Riemann invariants allows for similarity solutions that describe the shape of the outflow, as well as a theory that predicts properties of the Kelvin wave. A range of behaviours is observed, including flows that develop shocks and flows that continue to expand offshore. The qualitative behaviour of the outflow is strongly correlated with the value of a single dimensionless parameter that expresses the ratio of the speed of the flow driven by the Kelvin wave to that driven by image vorticity.

]]>The dynamics of a gas bubble in a square channel with a linearly increasing temperature at the walls in the vertical direction is investigated via three-dimensional numerical simulations. The channel contains a so-called ‘self-rewetting’ fluid whose surface tension exhibits a parabolic dependence on temperature with a well-defined minimum. The main objectives of the present study are to investigate the effect of Marangoni stresses on bubble rise in a self-rewetting fluid using a consistent model fully accounting for the tangential surface tension forces, and to highlight the effects of three-dimensionality on the bubble rise dynamics. In the case of isothermal and non-isothermal systems filled with a ‘linear’ fluid, the bubble moves in the upward direction in an almost vertical path. In contrast, strikingly different behaviours are observed when the channel is filled with a self-rewetting fluid. In this case, as the bubble crosses the location of minimum surface tension, the buoyancy-induced upward motion of the bubble is retarded by a thermocapillary-driven flow acting in the opposite direction, which in some situations, when thermocapillarity outweighs buoyancy, results in the migration of the bubble in the downward direction. In the later stages of this downward motion, as the bubble reaches the position of arrest, its vertical motion decelerates and the bubble encounters regions of horizontal temperature gradients, which ultimately lead to the bubble migration towards one of the channel walls. These phenomena are observed at sufficiently small Bond numbers (high surface tension). For stronger self-rewetting behaviour, the bubble undergoes spiralling motion. The mechanisms underlying these three-dimensional effects are elucidated by considering how the surface tension dependence on temperature affects the thermocapillary stresses in the flow. The effects of other dimensionless numbers, such as Reynolds and Froude numbers, are also investigated.

]]>The role of free-stream turbulence (FST) in the hydrodynamic instability mechanisms and transition to turbulence in laminar separation bubbles (LSBs) was investigated using direct numerical simulations (DNS). Towards this end, a set of highly resolved DNS have been carried out, where isotropic FST fluctuations with intensities from 0.1 % to 3 % are introduced to investigate the relevant physical mechanisms governing the interaction of separation and transition in LSBs. For disturbance-free simulations, i.e. without FST, laminar–turbulent transition involves a Kelvin–Helmholtz (KH) instability of the separated shear layer. For LSBs subjected to FST, vortical FST fluctuations penetrate the approaching attached laminar boundary layer upstream of the separation location and induce slowly growing low-frequency disturbances, so-called Klebanoff (K) modes, which cause a spanwise modulation with a distinct spanwise wavelength. Simultaneously, the FST enhances the initial levels of instability waves with frequencies in the frequency range of the KH instability, but at much smaller amplitude levels compared to the K-modes. Results from the calculations based on the linearized Navier–Stokes equations and comparison with DNS results reveal that the K-mode exhibits exponential growth in the separated shear layer until it reaches a peak amplitude. At the same time, two-dimensional (2D) disturbance waves are also exponentially amplified, in fact at larger growth rate compared to the K-mode, due to the primary (convective) shear-layer instability mechanism until they saturate downstream of the peak amplitude associated with the K-mode. Therefore, based on detailed spectral analysis and modal decompositions for the separation bubbles investigated, the transition process is the result of two different mechanisms: (i) strong amplification of high-frequency (order of the shedding frequency), essentially 2D or weakly oblique fluctuating disturbances and (ii) low-frequency, three-dimensional K-modes caused by FST. Depending on the intensity of the FST, one of these mechanisms would dominate the transition process, or both mechanisms act together and contribute simultaneously. The net effect of these two events is an acceleration of transition for an increased level of FST intensity, which in turn leads to a reduction of the extent of the separation bubble in streamwise and wall-normal directions. The ‘roll-up’ into spanwise large-scale vortical structures resulting from the shear-layer instability, and the eventual breakdown of these structures, strongly contribute to the reattachment process. The spanwise coherence of these ‘rollers’ deteriorates due to the presence of large-amplitude K-modes, thus effectively weakening their strength for high levels of FST intensities ( ).

]]>We study air entrainment by a solid plate plunging into a viscous liquid, theoretically and numerically. At dimensionless speeds of order unity, a near-cusp forms due to the presence of a moving contact line. The radius of curvature of the cusp’s tip scales with the slip length multiplied by an exponential of . The pressure from the air flow drawn inside the cusp leads to a bifurcation, at which air is entrained, i.e. there is ‘wetting failure’. We develop an analytical theory of the threshold to air entrainment, which predicts the critical capillary number to depend logarithmically on the viscosity ratio, with corrections coming from the slip in the gas phase.

]]>Experiments have shown that micron-sized distributed surface roughness can significantly promote transition in a three-dimensional boundary layer dominated by crossflow instability. This sensitive effect has not yet been fully explained physically and mathematically. Past studies focused on surface roughness exciting crossflow vortices and/or changing the local stability characteristics. The present paper seeks possible additional mechanisms by investigating the effects of distributed surface roughness on crossflow instability through resonant interactions with eigenmodes. A key observation is that the perturbation induced by roughness with specific wavenumbers can interact with two eigenmodes (travelling and stationary vortices) through triadic resonance, or interact with one eigenmode (stationary vortices) through Bragg scattering. Unlike the usual triadic resonance of neutral, or nearly neutral, eigenmodes, the present triadic resonance can take place among modes with growth rates, provided that these are equal; unlike the usual Bragg scattering involving neutral waves, crossflow stationary vortices can also be unstable. For these amplifying waves, the generalized triadic resonance and Bragg scattering are put forward, and the resulting corrections to the growth rates are derived by a multiple-scale method. The analysis is extended to the case where up to four crossflow vortices interact with each other in the presence of suitable roughness components. The numerical results for Falkner–Skan–Cooke boundary layers show that roughness with a small height (a few percent of the local boundary-layer thickness) can change growth rates substantially (by a more-or-less amount). This sensitive effect is attributed to two facts: (i) the resonant nature of the triadic interaction and Bragg scattering, which makes the correction to the growth rate proportional to the roughness height, and (ii) the wavenumbers of the roughness component required for the resonance are close to those of the neutral stationary crossflow modes, as a result of which a small roughness can generate a large response. Another important effect of roughness is that its presence renders the participating eigenmodes, which are otherwise independent, fully coupled. Our theoretical results suggest that micron-sized distributed surface roughness influences significantly both the amplification and spectral composition of crossflow vortices.

]]>The calving of icebergs accounts for a significant fraction of the mass loss from both the Antarctic and Greenland ice sheets. Iceberg melting affects the water properties impacting sea ice formation, local circulation and biological activity. Laboratory experiments have investigated the effects of the Earth’s rotation on iceberg melting and the possible formation of Taylor columns (TCs) underneath icebergs. It is found that at high Rossby number, , when rotation is weak compared to advection, iceberg melting is unaffected by the background rotation. As decreases, the melt rate shows an increasing trend, which is expected to reverse for very low . This behaviour is explained by considering the integrated horizontal velocity at the base of the iceberg. For moderate , a partial TC is formed and its effective blocking accelerates the flow under the remainder of the iceberg, which increases the melt rate since the melting is proportional to the flow velocity. It is expected that for very low the melt rate decreases because, with the expansion of the TC, the region of flow acceleration occurs away from the base of the iceberg. For low free stream velocity the freshwater produced by the ice melting introduces another dynamical effect. It is observed that there is a threshold free stream velocity below which the melt rate is constant. This is explained with the formation of a gravity current at the base of the iceberg that insulates it from the free flow and controls its melting.

]]>The goal of this paper is to identify and characterize the fundamental mechanisms that contribute toward flow stabilization in sheared plasma flows. Toward that end, we investigate the evolution of velocity and magnetic field perturbations in homogeneously sheared magnetohydrodynamic (MHD) flows subjected to an imposed streamwise magnetic field. The influences of magnetic field strength ( ) and perturbation wavevector orientation ( ) are characterized using linear analysis and direct numerical simulations. The linear analysis of ideal MHD indicates that the perturbation evolution is governed by four processes: pressure redistribution, kinetic energy production, kinetic–magnetic energy exchange and magnetic energy production due to magnetic stretching. The interplay between these processes can be characterized by the ratio of shear-to-magnetic timescales ( ) and , where , and are Alfvén wave speed, initial wavenumber and mean flow shear, respectively. For cases with low values of , a three-stage perturbation evolution is seen. At the first stage, pressure redistribution and production dominate leading to hydrodynamic-type behaviour. In the second stage, the onset of magnetic stretching process leads to an increase in magnetic energy. At late stages, production subsides and the dynamics is dominated by harmonic exchange between velocity and magnetic fields. For cases of , the magnetic field reacts rapidly enough that hydrodynamic and magnetic production stages occur simultaneously followed by harmonic exchange. In the case of , all three stages occur simultaneously leading to harmonic exchange between kinetic and magnetic energies without any perturbation growth. For all cases considered, the late stage harmonic exchange results in equipartition between perturbation magnetic and kinetic energies. For a given , the effect of increasing is to reduce the intensity of coupling and progressively slow down the three stages of evolution. For spanwise wavevector perturbations, the velocity–magnetic field interaction mechanism vanishes and there is no effect of pressure or magnetic field on individual velocity components.

]]>The contact line of a volatile liquid on a flat substrate is studied theoretically. We show that a remarkable result obtained for a pure-vapour atmosphere (Phys. Rev. E, vol. 87, 2013, 010401) also holds for an isothermal diffusion-limited vapour exchange with air. Namely, for both zero and finite Young’s angles, the motion- and phase-change-related contact-line singularities can in principle be regularised solely by the Kelvin effect (curvature dependence of saturation conditions). The latter prevents the curvature from diverging and rather leads to its versatile self-adjustment. To illustrate the point, the problem is resolved for a distinguished vicinity of the contact line (‘microregion’) in a ‘minimalist’ way, i.e. without any disjoining pressure, precursor film, Navier slip or any other microphysics. This also leads to the determination of the ‘Kelvin-only’ evaporation- and motion-induced apparent contact angles. With the Kelvin-only microscales actually turning out to be quite nanoscopic, other microphysics effects may nonetheless interfere too in reality. The Kelvin-only results will then yield a limiting case within such a more general formulation.

]]>Inviscid vortex models have been demonstrated to capture the essential physics of massively separated flows past aerodynamic surfaces, but they become computationally expensive as coherent vortex structures are formed and the wake is developed. In this work, we present a two-dimensional vortex model in which vortex sheets represent shear layers that separate from sharp edges of the body and point vortices represent the rolled-up cores of these shear layers and the other coherent vortices in the wake. We develop a circulation transfer procedure that enables each vortex sheet to feed its circulation into a point vortex instead of rolling up. This procedure reduces the number of computational elements required to capture the dynamics of vortex formation while eliminating the spurious force that manifests when transferring circulation between vortex elements. By tuning the rate at which the vortex sheets are siphoned into the point vortices, we can adjust the balance between the model’s dimensionality and dynamical richness, enabling it to span the entire taxonomy of inviscid vortex models. This hybrid model can capture the development and subsequent shedding of the starting vortices with insignificant wall-clock time and remain sufficiently low-dimensional to simulate long-time-horizon events such as periodic bluff-body shedding. We demonstrate the viability of the method by modelling the impulsive translation of a wing at various fixed angles of attack, pitch-up manoeuvres that linearly increase the angle of attack from to , and oscillatory pitching and heaving. We show that the proposed model correctly predicts the dynamics of large-scale vortical structures in the flow by comparing the distributions of vorticity and force responses from results of the proposed model with a model using only vortex sheets and, in some cases, high-fidelity viscous simulation.

]]>A discrepancy between the enhancement in overall burning rate and the enhancement in flame surface area measured for high-intensity turbulence is addressed. In order to reconcile the two quantities, an additional contribution from the effective turbulent diffusivity is considered. This contribution is expected to arise in sufficiently intense turbulence from eddies smaller than the flamelet thickness. In the present work, the enhancement in diffusivity arising from these eddies is estimated based on a model energy spectrum; individual contributions from all turbulence length scales smaller the flamelet thickness are integrated over the corresponding portion of the spectrum. It is shown that diffusivity enhancement, estimated in this manner, is able to account for the measured discrepancy between the overall burning rate enhancement and flame surface area enhancement. The factor quantifying this discrepancy is formalized as a closed-form function of the Karlovitz number.

]]>We investigate the effect of intergranular cohesive forces on the properties of self-diffusion in dense granular flows. The study is based on a series of simulated plane shear flows at different inertial and cohesion numbers, in which transverse diffusivities are measured. Results evidence an increase in diffusivity by up to two orders of magnitude when introducing cohesion. This strong effect is analysed using the Green–Kubo framework, expressing the diffusivity in terms of instantaneous grain velocity fluctuations and their time correlation. This analysis shows that cohesion, by forming enduring clusters in the flow, enhances the velocity fluctuations and their time persistence, which both contribute to enhancing grain mixing and self-diffusion.

]]>Transitional turbulence in shear flows is supported by a network of unstable exact invariant solutions of the Navier–Stokes equations. The network is interconnected by heteroclinic connections along which the turbulent trajectories evolve between invariant solutions. While many invariant solutions in the form of equilibria, travelling waves and periodic orbits have been identified, computing heteroclinic connections remains a challenge. We propose a variational method for computing orbits dynamically connecting small neighbourhoods around equilibrium solutions. Using local information on the dynamics linearized around these equilibria, we demonstrate that we can choose neighbourhoods such that the connecting orbits shadow heteroclinic connections. The proposed method allows one to approximate heteroclinic connections originating from states with multi-dimensional unstable manifold and thereby provides access to heteroclinic connections that cannot easily be identified using alternative shooting methods. For plane Couette flow, we demonstrate the method by recomputing three known connections and identifying six additional previously unknown orbits.

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