The self-similarity law for axisymmetric wakes has for the first time been examined and verified in a complex helical vortex in the far part of an asymmetric wake by means of direct numerical simulation (DNS). The helical vortex is the main coherent flow structure in the transitional non-axisymmetric wake behind an inclined 6:1 prolate spheroid at Reynolds number 3000 based on the minor axis. The gradual development of the complex helical vortex structure has been described in detail all the way from its inception at the spheroid and into the far wake. We observed a complex vortex composition in the generation stage, a rare jet-like wake pattern in the near wake and an abrupt change of helical symmetry in the vortex core without an accompanying change in flow topology, i.e. with no recirculation bubble.

Permeable and porous surfaces are common in natural and engineered systems. Flow and transport above such surfaces are significantly affected by the surface properties, e.g. matrix porosity and permeability. However, the relationship between such properties and macroscopic solute transport is largely unknown. In this work, we focus on mass transport in a two-dimensional channel with permeable porous walls under fully developed laminar flow conditions. By means of perturbation theory and asymptotic analysis, we derive the set of upscaled equations describing mass transport in the coupled channel–porous-matrix system and an analytical expression relating the dispersion coefficient with the properties of the surface, namely porosity and permeability. Our analysis shows that their impact on the dispersion coefficient strongly depends on the magnitude of the Péclet number, i.e. on the interplay between diffusive and advective mass transport. Additionally, we demonstrate different scaling behaviours of the dispersion coefficient for thin or thick porous matrices. Our analysis shows the possibility of controlling the dispersion coefficient, i.e. transverse mixing, by either active (i.e. changing the operating conditions) or passive mechanisms (i.e. controlling matrix effective properties) for a given Péclet number. By elucidating the impact of matrix porosity and permeability on solute transport, our upscaled model lays the foundation for the improved understanding, control and design of microporous coatings with targeted macroscopic transport features.

Streaks in pre-transitional boundary layers are analysed and their properties are extracted from direct numerical simulation data. Streaks that induce breakdown to turbulence via secondary instability are shown to differ from the remainder of the population in various attributes. Conditionally averaged flow fields establish that they are situated farther away from the wall, and have a larger cross-sectional area and higher peak amplitude. The analysis also shows that the momentum thickness acts as a similarity parameter for the properties of the streaks. Probability density functions of the streak amplitude, area, and shear along the streaks, collapse among the various pressure gradients when plotted as a function of the momentum thickness. A prediction scheme for laminar–turbulent transition based on artificial neural networks is presented, which can identify the streaks that will eventually induce the formation of turbulent spots. In comparison to linear stability theory, the approach achieves a higher prediction accuracy at considerably lower computational cost.

We study the propagation of viscous gravity currents over a thin porous substrate with finite capillary entry pressure. Near the origin, where the current is deep, propagation of the current coincides with leakage through the substrate. Near the nose of the current, where the current is thin and the fluid pressure is below the capillary entry pressure, drainage is absent. Consequently the flow can be characterised by the evolution of drainage and fluid fronts. We analyse this flow using numerical and analytical techniques combined with laboratory-scale experiments. At early times, we find that the position of both fronts evolve as $t^{1/2}$, similar to an axisymmetric gravity current on an impermeable substrate. At later times, the growing effect of drainage inhibits spreading, causing the drainage front to logarithmically approach a steady position. In contrast, the asymptotic propagation of the fluid front is quasi-self-similar, having identical structure to the solution of gravity currents on an impermeable substrate, only with slowly varying fluid flux. We benchmark these theoretical results with laboratory experiments that are consistent with our modelling assumption, but that also highlight the detailed dynamics of drainage inhibited by finite capillary pressure.

We investigate air entrainment and bubble statistics in three-dimensional breaking waves through novel direct numerical simulations of the two-phase air–water flow, resolving the length scales relevant for the bubble formation problem, the capillary length and the Hinze scale. The dissipation due to breaking is found to be in good agreement with previous experimental observations and inertial scaling arguments. The air entrainment properties and bubble size statistics are investigated for various initial characteristic wave slopes. For radii larger than the Hinze scale, the bubble size distribution, can be described by $N(r,t)=B(V_{0}/2{\rm\pi})({\it\varepsilon}(t-{\rm\Delta}{\it\tau})/Wg)r^{-10/3}r_{m}^{-2/3}$ during the active breaking stages, where ${\it\varepsilon}(t-{\rm\Delta}{\it\tau})$ is the time-dependent turbulent dissipation rate, with ${\rm\Delta}{\it\tau}$ the collapse time of the initial air pocket entrained by the breaking wave, $W$ a weighted vertical velocity of the bubble plume, $r_{m}$ the maximum bubble radius, $g$ gravity, $V_{0}$ the initial volume of air entrained, $r$ the bubble radius and $B$ a dimensionless constant. The active breaking time-averaged bubble size distribution is described by $\bar{N}(r)=B(1/2{\rm\pi})({\it\epsilon}_{l}L_{c}/Wg{\it\rho})r^{-10/3}r_{m}^{-2/3}$, where ${\it\epsilon}_{l}$ is the wave dissipation rate per unit length of breaking crest, ${\it\rho}$ the water density and $L_{c}$ the length of breaking crest. Finally, the averaged total volume of entrained air, $\bar{V}$, per breaking event can be simply related to ${\it\epsilon}_{l}$ by $\bar{V}=B({\it\epsilon}_{l}L_{c}/Wg{\it\rho})$, which leads to a relationship for a characteristic slope, $S$, of $\bar{V}\propto S^{5/2}$. We propose a phenomenological turbulent bubble break-up model based on earlier models and the balance between mechanical dissipation and work done against buoyancy forces. The model is consistent with the numerical results and existing experimental results.

The neutralization of a dilute spray of electrically charged droplets by ions of the opposite polarity generated by a corona discharge at a wire ring is analysed numerically. A Lagrangian description of the spray and Eulerian descriptions of the gas and the ions are used to deal with this two-way coupled problem. A model of the corona consisting of a line of charge and a distribution of ion sources is proposed. In the configuration that is analysed, neutralization usually begins at the shroud of the spray and extends to inner regions when the corona current increases. The number density of droplets is large at the shroud due to neutralized droplets that are no longer pushed by the electric field. These droplets can be dragged towards a collector surface by a weak forced flow that overcomes the ionic wind due to the force of the ions on the gas. The fraction of the spray charge that is neutralized increases with the corona current, but the value of this current required for full neutralization is several times larger than the inlet electric current of the spray owing to loss of ions to the boundaries of the system. The electric field induced by the charge of the droplets opposes the field due to the voltage applied between the wire ring and the extractor through which the droplets are injected, and thus reduces the threshold voltage of the corona and significantly affects its current–voltage characteristic, which may become multivalued. In turn, the electric field due to the applied voltage and the space charge of the ions affects the shape of the spray and the velocity of the droplets.

This paper presents a numerical study of vortex formation in the impulsively started viscous flow past an infinite wedge, for wedge angles ranging from $60^{\circ }$ to $150^{\circ }$. The Navier–Stokes equations are solved in the vorticity-streamfunction formulation using a time-splitting scheme. The vorticity convection is computed using a semi-Lagrangian method. The vorticity diffusion is computed using an implicit finite difference scheme, after mapping the physical domain conformally onto a rectangle. The results show details of the vorticity evolution and associated streamline and streakline patterns. In particular, a hierarchical formation of recirculating regions corresponding to alternating signs of vorticity is revealed. The appearance times of these vorticity regions of alternate signs, as well as their dependence on the wedge angles, are investigated. The scaling behaviour of the vortex centre trajectory and vorticity is reported, and solutions are compared with those available from laboratory experiments and the inviscid similarity theory.

We propose a two-phase two-thin-layer model for fluidized debris flows that takes into account dilatancy effects, based on the closure relation proposed by Roux & Radjai (Physics of Dry Granular Media, 1998, Springer, pp. 229–236). This relation implies that the occurrence of dilation or contraction of the granular material depends on whether the solid volume fraction is respectively higher or lower than a critical value. When dilation occurs, the fluid is sucked into the granular material, the pore pressure decreases and the friction force on the granular phase increases. On the contrary, in the case of contraction, the fluid is expelled from the mixture, the pore pressure increases and the friction force diminishes. To account for this transfer of fluid into and out of the mixture, a two-layer model is proposed with a fluid layer on top of the two-phase mixture layer. Mass and momentum conservation are satisfied for the two phases, and mass and momentum are transferred between the two layers. A thin-layer approximation is used to derive average equations, with accurate asymptotic expansions. Special attention is paid to the drag friction terms that are responsible for the transfer of momentum between the two phases and for the appearance of an excess pore pressure with respect to the hydrostatic pressure. For an appropriate form of dilatancy law we obtain a depth-averaged model with a dissipative energy balance in accordance with the corresponding three-dimensional initial system.

The vortex force is studied for a flat plate at arbitrarily large angle of attack. A suitable vortex force approach, adapted from a previous work, is used to study the vortex force and to build a vortex force line map to identify the force effect of any potential vortex. This map can be used exactly for a potential point vortex and approximately for a concentrated leading-edge vortex (LEV) or trailing-edge vortex (TEV); the latter are shown to have a non-potential vortex core. By means of this map, we identify a force-producing critical region, due to pressure suction, above the front and rear parts of the plate for an LEV and a TEV, respectively. The impulsively started flow problem is used as an application, with validation by computational fluid dynamics. The force variation in time is decomposed into four repeatable stages (force release, force enhancement, stall and force recovery) in close relation to the individual and combined effect by an LEV and a TEV. A pressure distribution analysis shows that force enhancement is due to pressure suction by an LEV, while stall and force recovery are respectively due to the upwash effect (which reduces the pressure below the plate) of a new TEV right off the plate and the pressure suction of this TEV having now moved above the plate. A viscous effect causes a small-amplitude oscillation on the force curves by promoting multiple small-scale LEVs.

The free fall of heavy bodies in a viscous fluid medium is a problem of interest to many engineering and scientific disciplines, including the study of unpowered flight and seed dispersal. The falling behaviour of coins and thin discs in particular has been categorized into one of four distinct modes; steady, fluttering, chaotic or tumbling, depending on the moment of inertia and Reynolds number. This paper investigates, through a carefully designed experiment, the falling dynamics of thin discs with central holes. The effects of the central hole on the disc’s motion is characterized for a range of Reynolds number, moments of inertia and inner to outer diameter ratio. By increasing this ratio, that is, the hole size, the disc is found to transition from tumbling to chaotic then fluttering at values of the moment of inertia not predicted by the falling modes of whole discs. This transition from tumbling to fluttering with increased hole size is viewed as a stabilization process. Flow visualization of the wake behind annular discs shows the presence of a vortex ring at the disc’s outer edge, as in the case of whole discs, and an additional counter-rotating vortex ring at the disc’s inner edge. The inner vortex ring is responsible for stabilizing the disc’s falling motion. These findings have significant implications on the development of design principles for engineered robotic systems in free flight, and may shed light on the stability of gliding animals.

We derive the John–Sclavounos equations, describing the motion of a fluid particle on the sea surface, from first principles using Lagrangian and Hamiltonian formalisms applied to the motion of a frictionless particle constrained on an unsteady surface. This framework leads to a number of new insights into the particle kinematics. The main result is that vorticity generated on a stress-free surface vanishes at a wave crest when the horizontal particle velocity equals the crest propagation speed, which is the kinematic criterion for wave breaking. If this holds for the largest crest, then the symplectic two-form associated with the Hamiltonian dynamics reduces instantaneously to that associated with the motion of a particle in free flight, as if the surface did not exist. Further, exploiting the conservation of the Hamiltonian function for steady surfaces and travelling waves, we show that particle velocities remain bounded at all times, ruling out the possibility of the finite-time blowup of solutions.

The aerodynamic performance of lifting surfaces operating at low Reynolds number conditions is impaired by laminar separation. In most cases, transition to turbulence occurs in the separated shear layer as a result of a series of strong hydrodynamic instability mechanisms. Although the understanding of these mechanisms has been significantly advanced over the past decades, key questions remain unanswered about the influence of external factors such as free-stream turbulence (FST) and others on transition and separation. The present study is driven by the need for more accurate predictions of separation and transition phenomena in ‘real world’ applications, where elevated levels of FST can play a significant role (e.g. turbomachinery). Numerical investigations have become an integral part in the effort to enhance our understanding of the intricate interactions between separation and transition. Due to the development of advanced numerical methods and the increase in the performance of supercomputers with parallel architecture, it has become feasible for low Reynolds number application ($O(10^{5})$) to carry out direct numerical simulations (DNS) such that all relevant spatial and temporal scales are resolved without the use of turbulence modelling. Because the employed high-order accurate DNS are characterized by very low levels of background noise, they lend themselves to transition research where the amplification of small disturbances, sometimes even growing from numerical round-off, can be examined in great detail. When comparing results from DNS and experiment, however, it is beneficial, if not necessary, to increase the background disturbance levels in the DNS to levels that are typical for the experiment. For the current work, a numerical model that emulates a realistic free-stream turbulent environment was adapted and implemented into an existing Navier–Stokes code based on a vorticity–velocity formulation. The role FST plays in the transition process was then investigated for a laminar separation bubble forming on a flat plate. FST was shown to cause the formation of the well-known Klebanoff mode that is represented by streamwise-elongated streaks inside the boundary layer. Increasing the FST levels led to accelerated transition, a reduction in bubble size and better agreement with the experiments. Moreover, the stage of linear disturbance growth due to the inviscid shear-layer instability was found to not be ‘bypassed’.

We report on the dynamics of circular finite-release Boussinesq gravity currents on a uniform slope. The study comprises a series of highly resolved direct numerical simulations for a range of slope angles between $5^{\circ }$ and $20^{\circ }$. The simulations were fixed at Reynolds number $Re=5000$ for all slopes considered. The temporal evolution of the front is compared to available experimental data. One of the interesting aspects of this study is the detection of a converging flow towards the centre of the gravity current. This converging flow is a result of the finite volume of the release coupled with the presence of a sloping boundary, which results in a second acceleration phase in the front velocity of the current. The details of the dynamics of this second acceleration and the redistribution of material in the current leading to its development will be discussed. These finite-release currents are invariably dominated by the head where most of the mixing and ambient entrainment occurs. We propose a simple method for defining the head of the current from which we extract various properties including the front Froude number and entrainment coefficient. The Froude number is seen to increase with steeper slopes, whereas the entrainment coefficient is observed to be weakly dependent on the bottom slope.

This paper presents three-dimensional (3D) direct numerical simulations (DNS) of flow past a circular cylinder over a range of Reynolds number ($Re$) up to 300. The gradual wake transition process from mode A* (i.e. mode A with large-scale vortex dislocations) to mode B is well captured over a range of $Re$ from 230 to 260. The mode swapping process is investigated in detail with the aid of numerical flow visualization. It is found that the mode B structures in the transition process are developed based on the streamwise vortices of mode A or A* which destabilize the braid shear layer region. For each case within the transition range, the transient mode swapping process consists of dislocation and non-dislocation cycles. With the increase of $Re$, it becomes more difficult to trigger dislocations from the pure mode A structure and form a dislocation cycle, and each dislocation stage becomes shorter in duration, resulting in a continuous decrease in the probability of occurrence of mode A* and a continuous increase in the probability of occurrence of mode B. The occurrence of mode A* results in a relatively strong flow three-dimensionality. A critical condition is confirmed at approximately $Re=265{-}270$, where the weakest flow three-dimensionality is observed, marking a transition from the disappearance of mode A* to the emergence of increasingly disordered mode B structures.

A small-amplitude sinusoidal surface undulation on the lower wall of Couette flow induces a vorticity perturbation. Using linear analysis, this vorticity field is examined when the fluid is viscoelastic and contrasted to the Newtonian configuration. For strongly elastic Oldroyd-B fluids, the penetration of induced vorticity into the bulk can be classified using two dimensionless quantities: the ratios of (i) the channel depth and of (ii) the shear-waves’ critical layer depth to the wavelength of the surface roughness. In the shallow-elastic regime, where the roughness wavelength is larger than the channel depth and the critical layer is outside of the domain, the bulk flow response is a distortion of the tensioned streamlines to match the surface topography, and a constant perturbation vorticity fills the channel. This vorticity is significantly amplified in a thin solvent boundary layer at the upper wall. In the deep-elastic case, the critical layer is far from the wall and the perturbation vorticity decays exponentially with height. In the third, transcritical regime, the critical layer height is within a wavelength of the lower wall and a kinematic amplification mechanism generates vorticity in its vicinity. The analysis is extended to localized, Gaussian wall bumps using Fourier synthesis. The Newtonian flow response consists of a single vortex above the bump. In the shallow-elastic flow, a second vortex with opposite circulation is established upstream of the surface protrusion and is induced by the vorticity layer on the upper wall. In the deep transcritical case, the perturbation field consists of a pair of counter-rotating vortices centred on the large vorticity around the critical layer. The more realistic FENE-P model, which accounts for the finite extensibility of the polymer chains, shows the same qualitative behaviour.

We present an investigation of rapidly rotating (small Rossby number $Ro\ll 1$) stratified turbulence where the stratification strength is varied from weak (large Froude number $Fr\gg 1$) to strong ($Fr\ll 1$). The investigation is set in the context of a reduced model derived from the Boussinesq equations that retains anisotropic inertia-gravity waves with order-one frequencies and highlights a regime of wave–eddy interactions. Numerical simulations of the reduced model are performed where energy is injected by a stochastic forcing of vertical velocity, which forces wave modes only. The simulations reveal two regimes: characterized by the presence of well-formed, persistent and thin turbulent layers of locally weakened stratification at small Froude numbers, and by the absence of layers at large Froude numbers. Both regimes are characterized by a large-scale barotropic dipole enclosed by small-scale turbulence. When the Reynolds number is not too large, a direct cascade of barotropic kinetic energy is observed, leading to total energy equilibration. We examine net energy exchanges that occur through vortex stretching and vertical buoyancy flux and diagnose the horizontal scales active in these exchanges. We find that the baroclinic motions inject energy directly to the largest scales of the barotropic mode, implying that the large-scale barotropic dipole is not the end result of an inverse cascade within the barotropic mode.

Dimensional analysis suggests that the dissipation length scale ($\ell _{{\it\epsilon}}=u_{\star }^{3}/{\it\epsilon}$) is the appropriate scale for the shear-production range of the second-order streamwise structure function in neutrally stratified turbulent shear flows near solid boundaries, including smooth- and rough-wall boundary layers and shear layers above canopies (e.g. crops, forests and cities). These flows have two major characteristics in common: (i) a single velocity scale, i.e. the friction velocity ($u_{\star }$) and (ii) the presence of large eddies that scale with an external length scale much larger than the local integral length scale. No assumptions are made about the local integral scale, which is shown to be proportional to $\ell _{{\it\epsilon}}$ for the scaling analysis to be consistent with Kolmogorov’s result for the inertial subrange. Here ${\it\epsilon}$ is the rate of dissipation of turbulent kinetic energy (TKE) that represents the rate of energy cascade in the inertial subrange. The scaling yields a log-law dependence of the second-order streamwise structure function on ($r/\ell _{{\it\epsilon}}$), where $r$ is the streamwise spatial separation. This scaling law is confirmed by large-eddy simulation (LES) results in the roughness sublayer above a model canopy, where the imbalance between local production and dissipation of TKE is much greater than in the inertial layer of wall turbulence and the local integral scale is affected by two external length scales. Parameters estimated for the log-law dependence on ($r/\ell _{{\it\epsilon}}$) are in reasonable agreement with those reported for the inertial layer of wall turbulence. This leads to two important conclusions. Firstly, the validity of the $\ell _{{\it\epsilon}}$-scaling is extended to shear flows with a much greater imbalance between production and dissipation, indicating possible universality of the shear-production range in flows near solid boundaries. Secondly, from a modelling perspective, $\ell _{{\it\epsilon}}$ is the appropriate scale to characterize turbulence in shear flows with multiple externally imposed length scales.

This work aims to provide a systematic experimental study of the wake behind two staggered square cylinders at a Reynolds number $Re=1.3\times 10^{4}$. Four distinct flow regimes, i.e. two single-street modes S-I and S-II and two double-street modes T-I and T-II, are identified based on extensive data, including Strouhal numbers $(St)$, flow structures and their downstream evolution. S-I, S-II and T-II are each further subdivided into two types. The flow characteristics in each regime are presented in terms of shear layer reattachment and impingement, vortex impingement, gap flow behaviour, interaction between cylinders and downstream evolution of the wake. A detailed discussion is made regarding how the physical aspects of the flow are connected to the initial conditions and the $St$ number.

This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number $Ro$, Froude number $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number $|Ro|$ is large and is stabilized for small and moderate $|Ro|$ when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber $m=1$ when the Burger number $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ is below a threshold depending on $Ro$. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small $Bu$. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.

The turbulent/non-turbulent interface is analysed in a direct numerical simulation of a boundary layer in the Reynolds number range $Re_{{\it\theta}}=2800{-}6600$, with emphasis on the behaviour of the relatively large-scale fractal intermittent region. This requires the introduction of a new definition of the distance between a point and a general surface, which is compared with the more usual vertical distance to the top of the layer. Interfaces are obtained by thresholding the enstrophy field and the magnitude of the rate-of-strain tensor, and it is concluded that, while the former are physically relevant features, the latter are not. By varying the threshold, a topological transition is identified as the interface moves from the free stream into the turbulent core. A vorticity scale is defined which collapses that transition for different Reynolds numbers, roughly equivalent to the root-mean-squared vorticity at the edge of the boundary layer. Conditionally averaged flow variables are analysed as functions of the new distance, both within and outside the interface. It is found that the interface contains a non-equilibrium layer whose thickness scales well with the Taylor microscale, enveloping a self-similar layer spanning a fixed fraction of the boundary-layer thickness. Interestingly, the straining structure of the flow is similar in both regions. Irrotational pockets within the turbulent core are also studied. They form a self-similar set whose size decreases with increasing depth, presumably due to breakup by the turbulence, but the rate of viscous diffusion is independent of the pocket size. The raw data used in the analysis are freely available from our web page (http://torroja.dmt.upm.es).