Proper simulation and modelling of geophysical flows is crucial to the study of numerical weather prediction, wind energy and many other applications. When simulating the atmospheric boundary layer, Coriolis forces act as a result of Earth’s rotation. The horizontal component of Earth’s rotation, which is often neglected, influences the balance of vertical momentum. The horizontal component results in systematic differences in the structure and statistics of stratified atmospheric boundary layers as a function of the direction of the geostrophic velocity. These differences are particularly relevant to atmospheric flows which include inhomogeneous roughness elements such as drag disks or wind turbines since the presence of these drag elements alters the balance between turbulent stresses and the Coriolis contributions in Reynolds stress budgets. Even at latitudes as high as $45^{\circ }$, changing the geostrophic wind velocity vector direction alone changes the magnitude of shear stress, and therefore vertical transport of kinetic energy, in the conventionally neutral atmospheric boundary layer up to $15\,\%$. As such, the boundary layer height, shear and veer profiles, surface friction velocity and other key features are affected by the direction of the geostrophic wind. The influence of the horizontal component of Earth’s rotation in stable nocturnal boundary layers depends on the strength of the stratification as there is a strong influence in the present study and a weak influence in the GEWEX Atmospheric Boundary Layer Study (GABLS) case. A model of the effect of the horizontal component on the boundary layer shear stress is also proposed and verified with the present simulations. While not studied here, the present observations are also relevant to the oceanic Ekman boundary layer.

The elastic analogue of the Landau–Levich dip-coating problem, in which a plate is withdrawn from a bath of fluid on whose surface lies a thin elastic sheet, is analysed for angle of withdrawal $\unicode[STIX]{x1D703}$ to the horizontal. The flow is controlled by the elasticity number, $El$, which is a measure of the relative importance of viscous and bending stresses, and $\unicode[STIX]{x1D703}$. The leading-order solution for small $El$ is a steady profile in which the thickness of the film on the plate is found to vary as $El^{3/4}/(1-\cos \unicode[STIX]{x1D703})^{5/8}$. This prediction is confirmed in the limit $\unicode[STIX]{x1D703}\ll 1$ by comparison with numerical simulation. Finally, the circumstances under which the assumption of a steady solution is no longer valid are discussed, and the time-dependent solution is described.

For the problem of horizontal convection the Nusselt number based on entropy production is bounded from above by $C\,Ra^{1/3}$ as the horizontal convective Rayleigh number $Ra\rightarrow \infty$ for some constant $C$ (Siggers et al., J. Fluid Mech., vol. 517, 2004, pp. 55–70). We re-examine the variational arguments leading to this ‘ultimate regime’ by using the Wentzel–Kramers–Brillouin method to solve the variational problem in the $Ra\rightarrow \infty$ limit and exhibiting solutions that achieve the ultimate $Ra^{1/3}$ scaling. As expected, the optimizing flows have a boundary layer of thickness ${\sim}Ra^{-1/3}$ pressed against the non-uniformly heated surface; but the variational solutions also have rapid oscillatory variation with wavelength ${\sim}Ra^{-1/3}$ along the wall. As a result of the exact solution of the variational problem, the constant $C$ is smaller than the previous estimate by a factor of $2.5$ for no-slip and $1.6$ for no-stress boundary conditions. This modest reduction in $C$ indicates that the inequalities used by Siggers et al. (J. Fluid Mech., vol. 517, 2004, pp. 55–70) are surprisingly accurate.

The Lagrangian mass transport by non-dissipating surface gravity wavepackets consists of the Stokes drift and the wave-induced return flow. We examine how directional spreading and density stratification affect this mass transport for an isolated non-dissipating wavepacket in deep water using a perturbation expansion. For an unstratified ocean, we show that the net displacement by the return flow is finite, negative, the same at all vertical levels and inversely proportional to the depth for spanwise-infinite packets representing unidirectional (two-dimensional) seas, but zero for spanwise-localised packets representing directionally spread seas (three-dimensional). We resolve this difference by demonstrating that a transition between two-dimensional-like (finite) and three-dimensional-like (zero) displacement occurs on a time scale inversely proportional to the degree of directional spreading. For a stratified ocean, we show that in two dimensions the net displacement profile by the return flow oscillates slowly with depth, with a wavelength dependent on the ratio of buoyancy frequency to the surface wave group velocity, and infinite displacements are predicted when the surface wavepacket resonantly excites internal waves. In three dimensions, the net displacement remains zero in the presence of stratification, but finite-time displacements may be appreciably altered.

In this paper, a fictitious domain method is used to perform interface-resolved numerical simulations of particle-laden turbulent flows in a vertical channel filled with Bingham fluids. The interactions between the particles and the turbulence are investigated at a bulk Reynolds number of 8000 (based on the mean velocity and the channel width), a particle/channel size ratio of 0.1 and a particle volume fraction of 2 %. The Bingham number based on the mean velocity and the channel half-width is 0.0, 0.54 and 3.0, respectively. Both neutrally buoyant and weak particle settling effects are considered. Our results indicate that the yield stress of Bingham fluids has drag-reduction effects by weakening the turbulence, and the addition of the particles enhances the flow friction. The Saffman lift force is important to the particle concentration distribution across the channel. Particle sedimentation attenuates the fluctuating velocity and Reynolds stress, resulting in a smaller flow friction compared with the neutrally buoyant case. When the fluid plasticity is strong, the single-phase flow has considerable temporal and spatial intermittencies. The vortex structures in one half-channel can be stronger than those in the other half-channel for a long period, resulting in the asymmetry of the statistics for a long time. The vortices are preferentially concentrated in streamwise-extended bands. The vortex bands and the laminar flow regions are largely segregated in the spanwise directions, and the vortex bands in the two half-channels are also typically staggered in the spanwise direction. The addition of the particles suppresses such vortex structures, the enormous intermittency and the asymmetry during the transition in the single-phase flow.

This paper presents a theoretical analysis on entropy wave generation in a heated one-dimensional duct, which is a simple thermoacoustic model of a combustor. Following a new observation that an entropy wave is caused by the fluctuations of heat/flow power ratio, the entropy transport equation is analytically solved for a heat input uniformly distributed over an acoustically compact zone. Investigating transfer functions from heat and acoustic fluctuations to the entropy wave, we obtain a deeper understanding on the low-pass filtering property of entropy waves with a closed-form expression of the entropic cutoff frequency. A theoretical explanation on why a thinner flame generally results in a stronger entropy wave is also given. These findings are extended to a general flame distribution from a temporal-spatial filtering interpretation of the entropy transport equation. Furthermore, the thin flame limits of our entropy wave models are compared with a popular entropy model based on the thin flame assumption and jump conditions. A numerical example supporting our theoretical findings is also presented.

The spatio-temporal dynamics of separation bubbles induced to form in a fully developed turbulent boundary layer (with Reynolds number based on momentum thickness of the boundary layer of 490) over a flat plate is studied via direct numerical simulations. Two different separation bubbles are examined: one induced by a suction–blowing velocity profile on the top boundary and the other by a suction-only velocity profile. The latter condition allows reattachment to occur without an externally imposed favourable pressure gradient and leads to a separation bubble more representative of those occurring over airfoils and in diffusers. The suction-only separation bubble exhibits a range of clearly distinguishable modes, including a high-frequency mode and a low-frequency ‘breathing’ mode that has been observed in some previous experiments. The high-frequency mode is well characterized by classical frequency scalings for a plane mixing layer and is associated with the formation and shedding of spanwise-oriented vortex rollers. The topology associated with the low-frequency motion is revealed by applying dynamic mode decomposition to the data from the simulations and is shown to be dominated by highly elongated structures in the streamwise direction. The possibility of Görtler instability induced by the streamwise curvature on the upstream end of the separation bubble as the underlying mechanism for these structures and the associated low frequency is explored.

We study the formation of fine radial jets during the impact of a compound drop on a smooth solid surface. The disperse-phase droplets are heavier than the outer continuous phase of the main drop and sink to the bottom of the drop before it is released from the nozzle. The droplets often arrange into a regular pattern around the axis of symmetry. This configuration produces narrow high-speed jets aligned with every internal droplet. These radial jets form during the early impulsive phase of the impact, by local focusing of the outer liquid, which is forced into the narrowing wedge under each internal droplet. The pressure-driven flow forces a thin sheet under and around each droplet, which levitates and separates from the solid surface. Subsequently, surface tension re-forms this horizontal sheet into a cylindrical jet, which is typically as narrow as ${\sim}35~\unicode[STIX]{x03BC}\text{m}$, while smaller droplets can produce even thinner jets. We systematically change the number of inner droplets and the properties of the main drop to identify the jetting threshold. The jet speed and thickness are minimally affected by the viscosity of the outer liquid, suggesting pure inertial focusing. The jets emerge at around eight times the drop impact velocity. Jetting stops when the density of the inner droplets approaches that of the continuous phase. The interior droplets are often greatly deformed and broken up into satellites by the outer viscous stretching, through capillary pinch-off or tip streaming.

The drag force on a sphere in tangential and normal motion to a plane wall is evaluated in the limit of large Knudsen number and small Mach (and Strouhal) number assuming isothermal conditions and diffuse reflection of gas molecules on walls. In the limit of free molecular flow, the molecular distribution function of the gas is evaluated using a set of coupled Fredholm integral equations. The results are compared with direct simulation Monte Carlo calculations and extended for finite Knudsen numbers. In all cases stronger dependence of the force on the width of the gap is found for normal compared to tangential motion. When the flow within the gap can be considered as essentially collisionless in nature, a similar dependence of the force on the gap width is observed at finite Knudsen numbers as in the free molecular case.

When the average intermolecular distance is comparable to the size of gas molecules, the Boltzmann equation, based on the dilute gas assumption, becomes invalid. The Enskog equation was developed to account for this finite size effect that makes the collision non-local and increases the collision frequency. However, it is time-consuming to solve the Enskog equation due to its complicated structure of collision operator and high dimensionality. In this work, on the basis of the Shakhov model, a gas kinetic model is proposed to simplify the Enskog equation for non-ideal monatomic dense gases. The accuracy of the proposed Shakhov–Enskog model is assessed by comparing its solutions of the normal shock wave structures with the results of the Enskog equation obtained by the fast spectral method. It is shown that the Shakhov–Enskog model is able to describe non-equilibrium flow of dense gases, when the maximum local mean free path of gas molecules is still greater than the size of a molecular diameter. The accuracy and efficiency of the present model enable simulations of non-equilibrium flow of dense gases for practical applications.

This paper discusses the generalised least-action principle and the associated concept of generalised flow introduced by Brenier (J. Am. Math. Soc., vol. 2, 1989, pp. 225–255), from the perspective of turbulence modelling. In essence, Brenier’s generalised least-action principle is a probabilistic generalisation of Arnold’s geometric interpretation of ideal fluid mechanics, whereby strong solutions to the Euler equations are deduced from minimising an action over Lagrangian maps. While Arnold’s framework relies on the deterministic concept of Lagrangian flow, Brenier’s least-action principle describes solutions to the Euler equations in terms of non-deterministic generalised flows, namely probability measures over sets of Lagrangian trajectories. In concept, generalised flows seem naturally fit to describe turbulent Lagrangian trajectories in terms of stochastic processes, an approach that originates from Richardson’s seminal work on turbulent dispersion. Still, Brenier’s generalised least-action principle has so far hardly found any practical application in the realm of fluid mechanics, let alone for turbulence modelling. The purpose of the present paper is to provide a hydrodynamical perspective on Brenier’s principle, and to assess its skills at coarse graining the Lagrangian motion of fluid particles, in order to reconstruct the space–time dynamics of the underlying Eulerian velocity fields. In practice, we rely on a statistical-mechanics interpretation of the concept of generalised flows, whereby the latter become akin to statistical ensembles of suitably defined ‘permutation flows’. We then employ Monte Carlo techniques to numerically solve the generalised least-action principle, and analyse the Eulerian and Lagrangian statistical features of the associated generalised flows. For simplicity, we restrict ourselves to two dimensions of space and consider situations of increasing complexity, ranging from solid rotation and cellular flows to freely decaying two-dimensional turbulence. Our analysis highlights a major caveat of the generalised variational principle. When used over long time lags, e.g. longer than a well-defined hydrodynamic turnover time, it generates artificial generalised flows, with non-physical statistical features. We argue that this limitation is not specifically inherent to Brenier’s formulation, but rather to the variational framework being formulated as a two-end boundary-value problem. When appropriately used over sufficiently short times, the generalised least-action principle is however relevant, and generalised flows can be explicitly constructed, that represent a space–time coarse graining of the underlying dynamics. We show numerical evidence that Brenier’s principle may then even accommodate irreversible Eulerian behaviours. This suggests that, if carefully used, generalised variational formulations could provide new tools to coarse grain genuine multi-scale hydrodynamics.

Stationary cross-flow vortex N-factors were calculated over the surface of a yawed circular cone using computationally predicted and experimentally observed wavenumber distributions. Surface heat-flux data were obtained on a $7^{\circ }$ half-angle circular cone to investigate the behaviour of the stationary waves at different angles of attack and Reynolds numbers at Mach 6 under quiet-flow conditions in the Boeing/AFOSR Mach-6 Quiet Tunnel at Purdue University. A wavelet analysis was conducted on the experimental surface heat-flux data to construct a spatial mapping of the local largest amplitude wavenumbers of the stationary cross-flow waves, which were between 40 and 80 per circumference. Significant axial and azimuthal variation was observed. The results from the wavelet analysis were used to inform the stability analysis. The computed integration marching directions demonstrated very good agreement with the experimentally observed paths. N-factors were first calculated by integrating the local amplification rate corresponding to the most amplified experimental wavenumbers. The calculations were repeated based on non-dimensional computationally varying wavenumber ratios, which were dimensionalized by the experimental data. The computed N-factors showed good agreement between the two techniques. N-factors were also computed using the computationally predicted most unstable wavenumbers. The results showed decreased agreement with the other two cases, suggesting that this assumption does not properly model the cross-flow transition process.

Viscous anti-parallel vortex reconnection is studied by means of direct numerical simulation for vortex Reynolds numbers $Re$ ($\equiv \unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D708}$, circulation/viscosity) up to 40 000. To suppress the inherent symmetry breaking due to the Kelvin–Helmholtz (planar jet) instability, as prevalent in prior studies, and to better explore the progression of the mechanism details, the simulation is performed by imposing symmetry and using double-precision arithmetic. We show, for the first time, the evidence of vortex reconnection cascade scenario initially proposed by Melander and Hussain (CTR Report, 1988), who suggested that the remnant threads, following the first reconnection, undergo successive reconnections in a cascade. Secondary reconnection (the details distinctly captured and visualized at a lower $Re=9000$) leads to the successive generation of numerous small-scale structures, including vortex rings, hairpin-like vortex packets and vortex tangles. As $Re$ increases, the third and higher generations of reconnection form a turbulent cloud avalanche consisting of a tangle of fine vortices. The energy is rapidly transferred to finer scales during reconnection, and a distinct - 5/3 inertial range is observed for the kinetic energy spectrum, associated with numerous resulting fine-scale bridgelets and thread filaments. In addition, we also discover an inverse cascade at large scales through the accumulation of bridgelets. The separation distance $\unicode[STIX]{x1D6FF}(t)$ before the first reconnection is found to scale as $t^{3/4}$, which is different from the typical 1/2 scaling for classical and quantum vortex filament reconnections. Both peak enstrophy and its production rate grow with $Re$ faster than the power law suggested by Hussain and Duraisamy (Phys. Fluids, vol. 23, 2011, 021701). Our simulations not only reveal the detailed mechanisms of high-$Re$ reconnection, but also shed light on the physics of turbulence cascade and present the reconnection avalanche as a realistic physical model for turbulence cascade.

We describe a three-dimensional (3-D) fluid–structure interaction (FSI) simulation study of the aortic valve, where the flow is driven by a specified transient pressure drop along the aorta tube. The thickness of the leaflets is varied from 0.05 mm to 0.8 mm so that the normalized bending rigidity by the systolic pressure gradient covers a wide range from $1.5\times 10^{-4}$ to 0.6. The non-uniform pressure distribution over the leaflets and the transient valve force are calculated, including the ‘water hammer’ effect during rapid closure. With low bending rigidity, the valve functions normally and produces physiological characteristics of healthy valves. However, exceedingly low rigidity leads to flapping motion of the leaflets and, in some cases (e.g. the normalized bending rigidity around 0.001), may reduce the performance index. As the leaflets become much stiffer, the valve is more difficult to open and slower to close, which leads to higher resistance and a reduced flow rate. Therefore, our results suggest that there is an optimal range of bending rigidity for the valve, roughly between 0.003 and 0.04 in normalized term. We further develop a one-dimensional unsteady flow model based on the momentum and mass conservation equations to replace the 3-D flow in the FSI simulation. The new flow model incorporates pressure loss across the valve as well as the leaflet motion. Comparison with the 3-D results shows that the reduced flow model is able to produce a reasonable 3-D deformation sequence of the leaflets, opening area and flow rate, especially in the cases of low bending rigidity.

Wake modes of a three-dimensional blunt-based body near a wall are investigated at a Reynolds number $Re=10^{5}$. The targeted modes are the static symmetry-breaking mode and two antisymmetric periodic modes. The static mode orientation is aligned with the horizontal major $y$-axis of the base and randomly switches between a positive $P$ and a negative $N$ state leading to long-time bistable dynamics of the turbulent wake. The modifications of these modes are studied when continuous blowing is applied at different locations through four slits along the base edges (denoted L for left, R for right, T for top and B for bottom) in either four single asymmetric configurations or two double symmetric configurations (denoted LR and TB). Two regimes, referred to as mass and momentum, are clearly identifiable for all configurations. The mass regime, which is fairly insensitive to blowing momentum and location, is characterized by the growth of the recirculating bubble as the total injected flow rate is increased, and is associated with a base drag reduction and interpreted as resulting from the equilibrium between mass fluxes feeding and emptying the recirculating region. A simple budget model is shown to be in agreement with entrainment velocities measured for isolated turbulent mixing layers. The strength of the static mode is reduced up to 20 % when the bubble length is maximum, whereas no change in the periodic mode frequencies is found. On the other hand, the momentum regime is characterized by the deflating of the recirculating bubble, leading to base drag increase, and it is interpreted by the free shear layer forcing, which increases the entrainment velocity, thus emptying the recirculating bubble. In this regime the static mode orientation is imposed by the blowing symmetry. Lateral L and R (respectively top/bottom T and B) blowing configurations select $P$ or $N$ states in the horizontal (respectively vertical) direction, while bistable dynamics persists for the symmetric LR and TB configurations. The shape of periodic modes follows the changes in wake static orientation. The transition between the two regimes is governed by both the total injected flow rate and the location of the injection.

We present a direct comparison between interface-resolved and one-way-coupled point-particle direct numerical simulations (DNS) of gravity-free turbulent channel flow laden with small inertial particles, with high particle-to-fluid density ratio and diameter of approximately three viscous units. The most dilute flow considered, solid volume fraction $O(10^{-5})$, shows the particle feedback on the flow to be negligible, whereas differences with respect to the unladen case, notably a drag increase of approximately 10 %, are found for a volume fraction $O(10^{-4})$. This is attributed to a dense layer of particles at the wall, caused by turbophoresis, flowing with large particle-to-fluid apparent slip velocity. The most dilute case is therefore taken as the benchmark for assessing the validity of a widely used point-particle model, where the particle dynamics results only from inertial and nonlinear drag forces. In the bulk of the channel, the first- and second-order moments of the particle velocity from the point-particle DNS agree well with those from the interface-resolved DNS. Close to the wall, however, most of the statistics show major qualitative differences. We show that this difference originates from the strong shear-induced lift force acting on the particles in the near-wall region. This mechanism is well captured by the lift force model due to Saffman (J. Fluid Mech., vol. 22 (2), 1965, pp. 385–400), while other widely used, more elaborate, approaches aiming at extending the lift model for a wider range of particle Reynolds numbers can actually underpredict the magnitude of the near-wall particle velocity fluctuations for the cases analysed here.

Experiments to study the effect of edge undulation on vortex formation have been conducted on impulsively accelerated plates. Abstractions of propulsors found in nature are produced by imprinting undulatory features with varying wavelengths onto the circumferential vortex-forming edge of circular plates. The effects of the small-scale disturbances introduced by these modifications are accessed by means of force measurements and time-resolved particle image velocimetry. Investigations of four different geometries at two different Reynolds numbers reveal an insensitivity of the flow towards length scales smaller than or similar to the thickness of the feeding shear layer. However, the instabilities in the shear layer and the coherence of the vortex wake are influenced when the wavelength of the undulation exceeds the shear-layer thickness by a significant margin. This results in a force augmentation due to enhanced entrainment into the turbulent vortex core, and thus an associated faster vortex growth rate. Yet, contrary to prior expectations, the time of vortex pinch-off remains constant for all edge modifications. The cause–effect relationship behind the stability of the vortex wake is further investigated. While for small edge undulations a turbulent transition of the vortex core results in vortex pinch-off, for larger edge undulations the turbulent vortex core is found to be fed constantly with additional circulation from the shear layer.

The general asymptotic theory for wave propagation in a slowly varying medium, classically known as the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation, is revisited here with the aim of constructing a new data diagnostic technique useful in atmospheric and oceanic sciences. Using the Wigner transform, a kind of mapping that associates a linear operator with a function, we analytically decompose a flow field into mutually independent wave signals. This method takes account of the variations in the polarisation relations, an eigenvector that represents the kinematic characteristics of each wave component, so as to project the variables onto their eigenspace quasi-locally. The temporal evolution of a specific mode signal obeys a single wave equation characterised by the dispersion relation that also incorporates the effect from the local gradient in the medium. Combining this method with transport theory and applying them to numerical simulation data, we can detect the transfer of energy or other conserved quantities associated with the propagation of each wave signal in a wide variety of situations.

Experiments of a turbulent mixing zone created by the Faraday instability at the statically stable interface between salt and fresh water are presented. The two-layer system, contained in a cuboidal tank of large dimensions, is accelerated vertically and periodically at various frequencies and amplitudes for three different density contrasts. We have developed a linear approach accounting for the full inhomogeneous and viscous problem, that is applied to a linear piecewise background density profile, and recovers the limiting cases of interface and homogeneous turbulence with a fully developed mixing layer. At onset, the wavelength of the most amplified modes and the corresponding Floquet exponent of the interface both verify our predictions. The dynamics is rather different when the instability is triggered from a sharp or diffuse interface: in the latter case, a change of characteristic wavelengths can be observed experimentally and explained by the theory. In the turbulent regime, the time evolution of the mixing zone size $L(t)$ for various experimental configurations compares well with confined direct numerical simulations. For some initial conditions, a short harmonic response of the instability is observed before the usual subharmonic one. Finally, the ultimate size of the mixing layer $L_{end}$, measured with a probe after the saturation of the instability and end of the forcing, is in excellent agreement with the recent theoretical prediction $L_{sat}=2{\mathcal{A}}g_{0}(2F+4)/\unicode[STIX]{x1D714}^{2}$, where $g_{0}$ is the gravitational acceleration, ${\mathcal{A}}$ the Atwood number, $\unicode[STIX]{x1D714}/2\unicode[STIX]{x03C0}$ the frequency and $F$ the acceleration ratio.

This paper describes an experimental investigation of the dynamics of freely falling thin circular disks settling through turbulence. The patterns of the three-dimensional disk motion are studied using an orthogonal arrangement of two high speed cameras. Turbulence is generated in a water tank using a random jet array facility where the jets are mounted in a co-planar configuration. The facility is run continuously until turbulence reaches a statistically stationary state, then, all water pumps are turned off simultaneously and a disk is released after a given waiting time. Contrary to spherical particles, finite-size inertial disks show an increase in the descent velocity for turbulence velocity fluctuations smaller than the particle descent velocity in quiescent flow. Thus, we observe a severe increase of the mean descent velocity of the disk with increasing magnitude of the turbulence velocity fluctuations (up to $20\,\%$ of the velocity in quiescent flow for the disk with higher dimensionless inertia $I^{\ast }$). We also observe descent events that do not exist for disks falling in still fluid; i.e. ‘slow tumbling’ events and ‘levitating’ events. Finally, we show that the dominant frequency of the particle oscillatory motion decreases for increasing descent velocity and that particles exhibit oscillatory frequencies that never exceed the dominant frequency in quiescent flow by more than $30\,\%$.