The fluid mechanics of hurricanes strongly depends on boundary layer energetics due to the warm-core nature of the system with peak velocities located at lower levels. One barrier that has inhibited a more complete characterization of energy transfer in the boundary layer is a lack of observations that resolve large, turbulent eddies. In particular, the occurrence and structure of upscale energy transfer (backscatter) in the hurricane boundary layer as well as the effects of backscatter on the vortex intensity are unknown. The analysis presented here of very high-resolution, three-dimensional wind observations from Hurricane Rita (2005) at peak intensity reveals large regions of organized backscatter in the boundary layer associated with coherent, turbulent eddies. Strong forwardscatter is also found next to the backscatter regions due to the interaction between adjacent eddies. Two components of the stress tensor are primarily responsible for this alternating scatter structure, as shown by large correlation coefficients between the fields: the radial–vertical component ($\tau _{13}$) and the azimuthal–vertical component ($\tau _{23}$) with average correlations of 79 % and 49 %, respectively. The Leonard, Reynolds and cross-term stress components are also provided. The impact of the sub-filter-scale energy transfer is estimated by computing the kinetic energy budget for the resolved-scale and eddy-scale motions. The results show that the sub-filter-scale energy transfer term is of the same order as the other terms in the eddy-scale budgets, contributing between 16 % and 40 % to the local time tendency with an average contribution of approximately 30 %. These results indicate that the coherent turbulent eddies can affect the vortex dynamics through wave–wave nonlinear interactions, which can subsequently influence the wave–mean flow interactions. This is the first study to examine the full sub-filter-scale energy transfer and its impact on the kinetic energy budget in the hurricane boundary layer. These findings emphasize the importance of coherent turbulence in the energy cascade and have the potential to improve turbulence closure schemes used in numerical simulations.

We propose a semi-analytical solution for the advection–diffusion equation in cylindrical domains, with an aim towards extracting blood flow rates from contrast variations in a coronary computed tomography angiography image. The solution proposed in this work, in contrast with existing methods, which only consider advection, incorporates both radial velocity variation and diffusion. By means of a Galerkin approach using Bessel functions, a solution for a three-dimensional concentration field at a single time point is obtained after a Laplace transformation. This semi-analytical solution forms the basis for a novel advection–diffusion flow estimation (ADFE) method. The ADFE is derived, validated through numerical spectral-element method computations, and shown to exhibit improved accuracy against the state-of-the-art method for image-based blood flow extraction.

We use direct numerical simulations to study the two-dimensional flow of a rotating, half soap bubble that is heated at its equator. The heating produces buoyancy and rotation generates Coriolis forces in the fluid. However, owing to the curved surface of the bubble, the buoyancy and Coriolis forces vary with latitude on the bubble, giving rise to rich flow behaviour. We first explore the single-point properties of the flow, including the Reynolds and Nusselt numbers, mean fields and Reynolds stresses, all as a function of latitude. For a given Rayleigh number, we observe a non-monotonic dependence on the Rossby number $Ro$, and large-scale mean circulations that are strongly influenced by rotation. We then consider quantities that reveal the multiscale nature of the flow, including spectra and spectral fluxes of kinetic and thermal energy, enstrophy and structure functions of velocity and temperature. The fluxes show that just as for non-buoyant two-dimensional turbulence on a flat surface, there is an upscale flux of kinetic energy at larger scales (fed by buoyancy injection of turbulent kinetic energy at smaller scales), and a downscale flux of enstrophy at smaller scales. The kinetic energy spectrum and velocity structure functions are well described by Bolgiano–Obukhov (BO) scaling at scales where the effects of rotation are weak. The temperature structure functions do not, however, satisfy BO scaling in general, owing to strong intermittency in the temperature field.

To mimic the unsteady vortex–wall interaction of animal propulsion in a canonical test case, a vorticity-annihilating boundary layer was examined through the spin-down of a vortex from solid-body rotation. A cylindrical, water-filled tank was rapidly stopped, and the decay of the vortex from solid-body rotation was observed by means of planar and stereo particle image velocimetry. High Reynolds-number ($Re$) measurements were achieved by combining a large-scale facility (diameter, $D=13\ \textrm {m}$) with a novel approach to reduce end-wall effects. The influence of the boundary-layer formation at the tank's bottom wall was minimised by introducing a saturated salt-water layer. The experimental efforts have allowed us to assess the $Re$ dependency of the laminar–turbulent transition of the vorticity-annihilating side-wall boundary layer at scales similar to large cetaceans. The scaling of the transition mechanism and its onset time were found to agree with predictions from linear stability analysis. Furthermore, the growth rate of the curved turbulent boundary layer was also in good agreement with an empirical scaling formulated in the literature for much smaller $Re$. Eventually, the scaling of vorticity annihilation was addressed. The earlier onset of transition at high $Re$ compensates for the reduced effects of viscosity, leading to similar vorticity annihilation rates during the early stages of the spin-down for a wide $Re$ range.

The mechanisms of destabilisation of the flow through soft-walled channels/tubes are qualitatively different from those in rigid-walled conduits. The stability depends on two dimensionless parameters, the Reynolds number $(\rho _f V_f h_f/ \mu _f)$ and $\varSigma = (\rho _f G h_f^{2}/\mu _f^{2})$, where $\rho _f$ and $\mu _f$ are the fluid density and viscosity, $h_f$ and $V_f$ are the fluid length and velocity scale and $G$ is the wall elasticity modulus. There is an instability at zero Reynolds number when the dimensionless parameter $\varGamma = (\mu _f V_f / h_f G)$ exceeds a critical value. The low-Reynolds-number instability of the Couette flow past a compliant surface is well understood, and has been confirmed in experiments, but that in a pressure-driven flow is not completely understood. Two modes of instability at high Reynolds number have been predicted: the inviscid mode with an internal viscous layer, for which the transition Reynolds number scales as ${Re}_t \propto \varSigma ^{1/2}$; and the wall mode instability with a viscous layer at the wall, for which ${Re}_t \propto \varSigma ^{3/4}$. The wall mode instability has been observed in experiments at Reynolds number as low as 300 in a soft-walled tube and as low as 100 in a channel with one compliant wall, though the scaling of the transition Reynolds number differs from the theoretical prediction due to substantial wall deformation. Though the flow after transition shares many of the characteristics of hard-wall turbulence, it differs in significant ways, suggesting that soft-wall turbulence is a separate class distinct from hard-wall turbulence.

Turbulent mixing layers in nature are often characterised by the presence of a mean shear and an unstable buoyancy gradient between two streams of different velocities. Depending on the relative strength of shear versus buoyancy, either the former or the latter may dominate the turbulence and mixing between the two streams. In this paper, we present a phenomenological theory that leads to the identification of two distinct turbulent regimes: an early regime, dominated by mean shear, and a later regime dominated by buoyancy. The main theoretical result consists of the identification of a cross-over timescale that distinguishes between the shear- and the buoyancy-dominated turbulence. This cross-over time depends on three large-scale constants of the flow, namely, the buoyancy difference, the velocity difference between the two streams and the gravitational acceleration. We validate our theory against direct numerical simulations of a temporal turbulent mixing layer compounded with an unstable stratification. We observe that the cross-over time correctly predicts the transition from shear- to buoyancy-driven turbulence, in terms of turbulent kinetic energy production, energy spectra scaling and mixing layer thickness.

The local streamline topology classification method of Chong et al. (Phys. Fluids A: Fluid Dyn., vol. 2, no. 5, 1990, pp. 765–777) is adapted and extended to describe the geometry of infinitesimal vortex lines. Direct numerical simulation (DNS) data of forced isotropic turbulence reveals that the joint probability density function (p.d.f.) of the second ($q_\omega$) and third ($r_\omega$) normalized invariants of the vorticity gradient tensor asymptotes to a self-similar bell shape for $Re_\lambda > 200$. The same p.d.f. shape is also seen at the late stages of breakdown of a Taylor–Green vortex suggesting the universality of the bell-shaped p.d.f. form in turbulent flows. Additionally, vortex reconnection from different initial configurations is examined. The local topology and geometry of the reconnection bridge is shown to be nearly identical in all cases considered in this work. Overall, topological characterization of the vorticity field provides a useful analytical basis for examining vorticity dynamics in turbulence and other fluid flows.

The statistical relation between ocean wave geometry and water particle movements can be formulated in the stochastic Gauss–Lagrange model. In this paper we use Slepian models to obtain detailed information of the sea surface elevation in the neighbourhood of local maxima in a Gaussian wave model and of the movements of the top particle of the waves. We present full conditional distributions of the Gaussian vertical and horizontal movements in the Gauss–Lagrange model, and represent them as one regression component depending on the height and curvature at the wave maxima and one residual component. These conditional distributions define the explicit vertical and horizontal Slepian models. The Slepian models are used to simulate individual min–max–min waves in space, in particular their front–back asymmetry, and the velocity vector of the particle at the wave maximum. We find that there is a strong relation between the degree of front–back wave asymmetry and the direction of the particle movement. We discuss the role of second-order corrections to the Gaussian components and find only minor effects for the sea states studied. The Slepian model is shown to be an efficient tool to obtain detailed information about Gaussian and related models in the neighbourhood of critical points, without the need for time and space consuming simulations. In particular, they permit easy simulation of shape and kinematics of rare extreme waves.

This paper characterises the steady and time-periodic behaviour of swirling jets using numerical bifurcation analysis. Its objective is to elucidate the dynamics of fully developed, unconfined, laminar swirling jets under variations in the Reynolds number $\textit {Re}$ and swirl ratio $S$. Within the $(0,0)\leq (\textit {Re},S)\leq (300,3)$ range, the steady, axisymmetric flow exhibits several distinct patterns ranging from a quasi-columnar jet along the central axis at low $S$ to a radial jet attached to the containing wall at high $S$ with various forms of vortex breakdown in between. A cusp bifurcation appears in the steady solution manifold which triggers bistable behaviour due to a competition between inner and outer low pressure regions associated with vortex breakdown and entrainment of the ambient fluid, respectively. Instability of the steady flow is linked to eigenmodes which are singly ($|m|=1$) or doubly ($|m|=2$) azimuthally periodic, although additional instabilities with other azimuthal wavenumbers occur at $(\textit {Re},S)$-values beyond the leading neutral curves. The various branches of limit cycle solutions stemming from these neutral curves are associated with both super- and sub-critical Hopf bifurcations. The resulting unsteady flow fields exhibit a wide array of rotating, three-dimensional flow structures, and comparisons between the time-averaged and steady flow patterns highlight the role of these unsteady nonlinear interactions on the overall behaviour of swirling jets. Similarities and differences between this laterally unconfined jet and broader classes of swirling flows, including confined swirling jets and unconfined vortex models, are also discussed.

We present a comprehensive study of the mesoscale eddy forcing in the ocean by proposing spatially local filtering of the high-resolution double-gyre ocean circulation solution into its large- and small-scale (eddy) components. The large-scale component is dominated by the mid-latitude gyres, the western boundary currents and their highly transient eastward jet extension; the eddy component is concentrated around the eastward jet and strongly interacts with it. The proposed decomposition method achieves flow filtering based on the local spatial correlations. This is different from the existing decomposition methods, e.g. classical Reynolds decomposition and moving-average (spatial) filtering with a constant filter size based on the first baroclinic Rossby deformation radius. Next, we characterize the dynamical impacts of the resulting eddy forcing on the large-scale flow in terms of their mutual time-lagged spatial correlations, formulated as product integral characteristics. Its temporal statistics uncover robust causality between the eddy forcing and the induced large-scale potential vorticity anomalies – referred to as the eddy backscatter. The results also prove the significance of the transient eddy forcing and the time lag dependence of the eddy backscatter. We argue that these properties are to be considered by eddy parametrization schemes. We further used the decomposed eddy fields to augment a coarse-resolution ocean model. The augmented solution statistically reproduces the missing eastward jet extension, enhances the eddy activities around it and recovers the essential large-scale low-frequency variability. This justifies a reduced-order statistical emulation of the eddies – an emerging methodology for including eddy effects in non-eddy-resolving ocean models.

Inter-species drag forces in granular flows play a central role in setting the speed and extent of segregation, a process that separates grains of different size or density. Here, we study this drag force in detail, using a novel configuration of discrete element simulations that allows us to completely characterise the drag in inertial granular flows by studying it in a uniform environment. By applying opposing forces to grains in monodisperse and size-bidisperse shear flows, we show that the strength of the drag force scales as $I^{-7/4}$, where $I$ is the granular inertial number, and propose a model that explains this scaling by relating the strength of drag to grain velocity fluctuations. These findings suggest that much of the previously observed dependence of the segregation rate on the local shear rate and pressure in dense free-surface flows is due to variation in the strength of the inter-species drag, rather than the strength of forces that drive segregation.

Viscoelastic flow through an abrupt planar contraction geometry above a certain Weissenberg number ($Wi$) is well known to become unstable upstream of the contraction plane via a central jet separating from the walls and forming vortices in the salient corners. Here, for the first time, we consider three-dimensional (3-D) viscoelastic contraction flows in a microfabricated glass square–square contraction geometry. We employ state-of-the-art microtomographic particle image velocimetry to produce time-resolved and volumetric quantification of the 3-D viscoelastic instabilities arising in a dilute polymer solution driven through the geometry over a wide range of $Wi$ but at negligible Reynolds number. Based on our observations, we describe new insights into the growth, propagation and transient dynamics of an elastic vortex formed upstream of the 3-D microcontraction due to flow jetting towards the contraction. At low $Wi$ we observe vortex growth for increasing $Wi$, followed by a previously unreported vortex growth plateau region. In the plateau region, the vortex circulates around the jet with a period that decreases with $Wi$ but an amplitude that is independent of $Wi$. In addition, we report new out-of-plane asymmetric jetting behaviour with a phase-wise dependence on $Wi$. Finally, we resolve the rate-of-strain tensor $\boldsymbol{\mathsf{D}}$ and ascribe local gradients in $\boldsymbol{\mathsf{D}}$ as the underlying driver of circulation via strain hardening of the fluid in the wake of the jet.

This study investigates water waves generated by a bottom obstacle translating at a subcritical speed in constant water depth, using a combination of analytical and numerical approaches. The newly derived analytical solutions reveal two types of waves – the transient free waves that propagate radially outwards, and the trapped wave that stays on top of the translating bottom obstacle. Closed-form asymptotic solutions for both the free surface and the flow velocities are derived in the far field, and near the leading wave or in the shallow water limit. The far-field leading waves are mathematically shown to be insensitive to the exact shape of the obstacle. Numerical long-wave models are employed to examine effects unaccountable by the linear analytical solutions. Nonlinear effects are found to cause only small deviations from the linear solutions. The effects of the obstacle's acceleration and deceleration are also examined numerically. Overall, the idealised linear analytical solutions predict well the characteristics of water waves generated by a bottom obstacle, and therefore can serve as the cornerstone of a theory-based model for quickly predicting the tsunamis generated by a submarine landslide.

Hypersonic flows over a compression ramp with different leading-edge radii are numerically investigated. Flow separation occurs on the cold-wall compression ramp with a free-stream Mach number of 7.7 and a unit Reynolds number of $4.2 \times 10^6$ m$^{-1}$. By performing direct numerical simulations (DNS), it is shown that the separation bubble enlarges when the leading-edge radius is increased from zero up to a critical value. Beyond the critical radius, the separation bubble conversely shrinks as the radius is further increased. Global stability analysis (GSA) is employed to investigate the three-dimensional instability of the two-dimensional base flows. It is found that the inherent instability in the flow field also exhibits a reversal trend, that is, the flow system firstly becomes more unstable and then tends to be more stable with increasing leading-edge radius. The growth rate and spanwise wavelength of the unstable modes identified by GSA are verified by DNS. Accompanying the occurrence of three-dimensionality, streamwise heat-flux streaks are formed on the ramp surface downstream of reattachment. The present study demonstrates that a proper blunting of the leading edge can suppress flow separation, reduce aerodynamic heating and stabilise the flow system for a hypersonic compression-ramp flow.

Interfacial stability is important for many processes involving heat and mass transfer across two immiscible phases. For the evaporation of a binary solution with one component more volatile than the other, gradients in surface tension can arise. These gradients can ultimately destabilize the liquid–gas interface. We study the evaporation of an ethanol–water solution, for which ethanol is more volatile. The solution is contained in a horizontal Hele-Shaw cell which is open from one end to allow for evaporation into air. A Marangoni instability is then triggered at the liquid–air interface. We study the temporal evolution of the instability through its effects on the bulk of the liquid. More specifically, the growth of convective cells is measured with confocal microscopy and the velocity field with microparticle image velocimetry. The results of numerical simulations based on quasi-two-dimensional equations satisfactorily compare with the experimental observations, even without consideration of evaporative cooling, although this cooling can play an extra role in experiments. Furthermore, a linear stability analysis applied to a simplified version of the quasi-two-dimensional equations showed reasonably good agreement with the results from simulations at early times, when the instability has just been triggered and before coarsening. In particular, we find a critical Marangoni number below which a regime of stability is predicted.

We consider the dynamics of a two-dimensional incompressible perfect fluid on a Möbius strip embedded in $\mathbb {R}^{3}$. The vorticity–stream function formulation of the Euler equations is derived from an exterior-calculus form of the momentum equation. The non-orientability of the Möbius strip and the distinction between forms and pseudo-forms this introduces lead to unusual properties: a boundary condition is provided by the conservation of circulation along the single boundary of the strip, and there is no integral conservation for the vorticity density or for any odd function thereof. A finite-difference numerical implementation is used to illustrate the Möbius-strip realisation of familiar phenomena: translation of vortices along boundaries, shear instability and decaying turbulence.

This article presents a modelling approach to predict the low-frequency sound generated by entropy fluctuations interacting with isolated aerofoils. A model of the acoustic field is obtained based on a linearisation of the compressible Euler equations about a steady, potential, compressible mean flow. Mean flow variations of velocity and density are accounted for in the source term, but are neglected in the sound propagation. Using a Lorentz-type transformation, the problem is reduced to solving a Helmholtz equation. This equation is recast in integral form and a solution is obtained using a compact Green's function method. This approach places no restrictions on the entropy wavelength, while assuming that the acoustic wavelength is large compared to the profile chord and spacing. The source term is further simplified by assuming that the steady flow is a small perturbation to a uniform flow. The model is illustrated using a symmetric aerofoil and its performance is assessed against numerical simulations of the compressible Euler equations. Good agreement is found for all the frequencies of validity of the theory and for all the range of subsonic Mach numbers. The solution for a symmetric aerofoil interacting with plane entropy waves corresponds to the combination of a dipole along the horizontal axis and a monopole. The dipole originates from the unsteady drag experienced by the aerofoil owing to the fluctuations of density and the monopole from the strong local acceleration of the flow at the leading edge. The monopole term becomes negligible for low Mach numbers.

Microfluidic systems consisting of a square microchannel with an orthogonal side branch are promising tools to enrich or sort suspensions of deformable capsules. To allow their operating control, we numerically consider a train of initially spherical identical capsules, equally spaced along the axis of the feeding channel. The capsules have a strain-hardening membrane, an internal fluid viscosity identical to that of the external fluid and a size comparable to that of the channel. We study the influence of the interspacing on the capsule path selection at the channel bifurcation using a three-dimensional immersed boundary–lattice Boltzmann method. Our objectives are to establish a phase diagram and identify the critical interspacing above which hydrodynamic interaction between capsules no longer affects their path selection. We find two main regimes. At low interspacing, strong capsule interaction leads to an unsteady regime for which the capsule path selection follows either a periodic or a disordered state. Above a critical initial interspacing $d_{ct}$, a steady regime is achieved where interaction between capsules is too weak to affect their path selection. The capsules then follow an identical steady trajectory. We find that the dependence of the interspacing $d_{ct}$, normalised by the capsule radius, on the flow split ratio falls onto a universal curve regardless of the flow strength, capsule size and membrane shear elasticity. We also compare the path selection of a capsule train with that of a two-capsule system, and discuss applications of the present results in controlling capsule trains in microfluidic suspension enrichment devices.

Three-dimensional direct numerical simulations are used to characterize turbulent buoyant convection in a box-shaped Rayleigh–Bénard cavity with a rough bottom plate made of a series of square based blocks separated by valleys. The cavity is filled with water. The Rayleigh number varies over five decades up to $10^{10}$. As mentioned in the literature, three successive heat transfer regimes are identified: from inactive roughness (I) to a regime (III) where the heat transfer increase is larger than that expected from only surface increase due to roughness. The heat transfers of the transitional regime II are particularly intense. After validation against experimental and numerical data from the literature, we highlight the role of the fluid retained within valleys (the inner fluid). It is shown that only the heat transfer across the fluid interface between the cavity bulk and the inner fluid is responsible for changes in the overall heat transfer at the rough plate, with an exponent of the heat transfer scaling law close to $1/2$ in regime II. The valley flow typifies the limits of this regime: the blocks protrude from the thermal boundary layer while remaining within the kinetic boundary layer. As compared with regimes I and III, regime II is characterized by larger temperature fluctuations, especially near the rough plate, and a larger friction coefficient. A fluctuating rough fluid layer overlaying both blocks and valleys appears in regime III, in addition to the classic boundary layers formed along the plate geometry.

With a view to numerical modelling and optimisation of wave energy farms, a simple recursive formulation is employed to solve for the reflection and transmission of plane water waves by a number of rows of vertical obstacles, under the wide-spacing approximation. The proposed recursive formulation relies on the ‘concatenation’ of any two sets of obstacles, for which the reflection–transmission problem is already resolved. Furthermore, the obstacles are allowed to move in any combination of pitch and surge. The proposed recursive model is validated by means of physical experiments in a small-scale wave flume, whereby waves are reflected and transmitted by one, two and three rows of vertical, flexible blades, taking into account dissipation within the fluid along the wave propagation direction. For the special case of identical, regularly spaced rows, under the adopted formalism, distinct theoretical behaviours are highlighted, depending on whether or not individual obstacles absorb (or dissipate) energy as they interact with incoming waves. In a ‘non-dissipative’ case, the well known fact that discrete values of the row-to-row distance $L$ completely cancel reflection is retrieved, as well as the existence of ‘band-gap’ intervals, i.e. intervals for $L$ where reflection is high, with maximum reflection occurring away from the Bragg condition. In contrast, when the obstacles dissipate or absorb energy as they interact with the fluid, reflection is always non-zero, and, as the number of rows tends to infinity, forms marked Bragg peaks, reaching unity when $L$ is a multiple of half a wavelength.