The understanding of chaotic dynamics in high-dimensional systems that has emerged in the last decade offers a promising dynamical framework to study turbulence. Here turbulence is viewed as a walk through a forest of exact solutions in the infinite-dimensional state space of the governing equations. Recently, Chandler & Kerswell (J. Fluid Mech., vol. 722, 2013, pp. 554–595) carry out the most exhaustive study of this programme undertaken so far in fluid dynamics, a feat that requires every tool in the dynamicist’s toolbox: numerical searches for recurrent flows, computation of their stability, their symmetry classification, and estimating from these solutions statistical averages over the turbulent flow. In the long run this research promises to develop a quantitative, predictive description of moderate-Reynolds-number turbulence, and to use this description to control flows and explain their statistics.

We discuss the path of a tracer particle as a microswimmer moves past on an infinite, straight trajectory. If the tracer is sufficiently far from the path of the swimmer it moves in a closed loop. As the initial distance between the tracer and the path of the swimmer $\rho $ decreases, the tracer is displaced a small distance backwards (relative to the direction of the swimmer velocity). For much smaller tracer–swimmer separations, however, the tracer displacement becomes positive and diverges as $\rho \rightarrow 0$. To quantify this behaviour we calculate the Darwin drift, the total volume swept out by a material sheet of tracers, initially perpendicular to the swimmer path, during the swimmer motion. We find that the drift can be written as the sum of a universal term which depends on the quadrupolar flow field of the swimmer, together with a non-universal contribution given by the sum of the volumes of the swimmer and its wake. The formula is compared to exact results for the squirmer model and to numerical calculations for a more realistic model swimmer.

We present experimental results on high-Reynolds-number motion of partially non-wetting liquid drops on inclined plane surfaces using: (i) water on fluoro-alkyl silane (FAS)-coated glass; and (ii) mercury on glass. The former is a high-hysteresis ($3{5}^{\circ } $) surface while the latter is a low-hysteresis one (${6}^{\circ } $). The water drop experiments have been conducted for capillary numbers $0. 0003\lt Ca\lt 0. 0075$ and for Reynolds numbers based on drop diameter $137\lt Re\lt 3142$. The ranges for mercury on glass experiments are $0. 0002\lt Ca\lt 0. 0023$ and $3037\lt Re\lt 20\hspace{0.167em} 069$. It is shown that when $Re\gg 1{0}^{3} $ for water and $Re\gg 10$ for mercury, a boundary layer flow model accounts for the observed velocities. A general expression for the dimensionless velocity of the drop, covering the whole $Re$ range, is derived, which scales with the modified Bond number ($B{o}_{m} $). This expression shows that at low $Re$, $Ca\sim B{o}_{m} $ and at large $Re$, $Ca \sqrt{Re} \sim B{o}_{m} $. The dynamic contact angle (${\theta }_{d} $) variation scales, at least to first-order, with $Ca$; the contact angle variation in water, corrected for the hysteresis, collapses onto the low-$Re$ data of LeGrand, Daerr & Limat (J. Fluid Mech., vol. 541, 2005, pp. 293–315). The receding contact angle variation of mercury has a slope very different from that in water, but the variation is practically linear with $Ca$. We compare our dynamic contact angle data to several models available in the literature. Most models can describe the data of LeGrand et al. (2005) for high-viscosity silicon oil, but often need unexpected values of parameters to describe our water and mercury data. In particular, a purely hydrodynamic description requires unphysically small values of slip length, while the molecular-kinetic model shows asymmetry between the wetting and dewetting, which is quite strong for mercury. The model by Shikhmurzaev (Intl J. Multiphase Flow, vol. 19, 1993, pp. 589–610) is able to group the data for the three fluids around a single curve, thereby restoring a certain symmetry, by using two adjustable parameters that have reasonable values. At larger velocities, the mercury drops undergo a change at the rear from an oval to a corner shape when viewed from above; the corner transition occurs at a finite receding contact angle. Water drops do not show such a clear transition from oval to corner shape. Instead, a direct transition from an oval shape to a rivulet appears to occur.

Wind blowing over the ocean surface can be treated as a turbulent boundary layer over a multiscale rough surface with moving roughness elements, the waves. Large-eddy simulation (LES) of such flows is challenging because LES resolves wind–wave interactions only down to the grid scale, $\Delta $, while the effects of subgrid-scale (SGS) waves on the wind need to be modelled. Usually, a surface-layer model based on the law of the wall is used; but the surface roughness has been known to depend on the local wind and wave conditions and is difficult to parameterize. In this study, a dynamic model for the SGS sea-surface roughness is developed, with the roughness corresponding to the SGS waves expressed as ${\alpha }_{w} \hspace{0.167em} { \sigma }_{\eta }^{\Delta } $. Here, ${ \sigma }_{\eta }^{\Delta } $ is the effective amplitude of the SGS waves, modelled as a weighted integral of the SGS wave spectrum based on the geometric and kinematic properties of the waves for which five candidate expressions are examined. Moreover, ${\alpha }_{w} $ is an unknown dimensionless model coefficient determined dynamically based on the first-principles constraint that the total surface drag force or average surface stress must be independent of the LES filter scale $\Delta $. The feasibility and consistency of the dynamic sea-surface roughness models are assessed by a priori tests using data from high-resolution LES with near-surface resolution, appropriately filtered. Also, these data are used for a posteriori tests of the dynamic sea-surface roughness models in LES with near-surface modelling. It is found that the dynamic modelling approach can successfully capture the effects of SGS waves on the wind turbulence without ad hoc prescription of the model parameter ${\alpha }_{w} $. Also, for ${ \sigma }_{\eta }^{\Delta } $, a model based on the kinematics of wind–wave relative motion achieves the best performance among the five candidate models.

We demonstrate the existence of an exact invariant solution to the Navier–Stokes equations for the asymptotic suction boundary layer. The identified periodic orbit with a very long period of several thousand advective time units is found as a local dynamical attractor embedded in the stability boundary between laminar and turbulent dynamics. Its dynamics captures both the interplay of downstream-oriented vortex pairs and streaks observed in numerous shear flows as well as the energetic bursting that is characteristic for boundary layers. By embedding the flow into a family of flows that interpolates between plane Couette flow and the boundary layer, we demonstrate that the periodic orbit emerges in a saddle–node infinite-period (SNIPER) bifurcation of two symmetry-related travelling-wave solutions of plane Couette flow. Physically, the long period is due to a slow streak instability, which leads to a violent breakup of a streak associated with the bursting and the reformation of the streak at a different spanwise location. We show that the orbit is structurally stable when varying both the Reynolds number and the domain size.

The construction of approximate Schrödinger eigenvalue equations for planetary (Rossby) waves and for inertia–gravity (Poincaré) waves on an ocean-covered rotating sphere yields highly accurate estimates of the phase speeds and meridional variation of these waves. The results are applicable to fast rotating spheres such as Earth where the speed of barotropic gravity waves is smaller than twice the tangential speed on the equator of the rotating sphere. The implication of these new results is that the phase speed of Rossby waves in a barotropic ocean that covers an Earth-like planet is independent of the speed of gravity waves for sufficiently large zonal wavenumber and (meridional) mode number. For Poincaré waves our results demonstrate that the dispersion relation is linear, (so the waves are non-dispersive and the phase speed is independent of the wavenumber), except when the zonal wavenumber and the (meridional) mode number are both near 1.

Two-dimensional turbulent free-surface flow is considered. The ensemble-averaged flow quantities may depend on time. The slope of the plane bottom of the channel is assumed to be small. The roughness of the bottom is allowed to vary with the space coordinate, leading to small variations in the bottom friction coefficient. An asymptotic analysis, which is free of turbulence modelling, is performed for large Reynolds numbers and Froude numbers close to the critical value 1. As a result, an extended Korteweg–deVries (KdV) equation for the surface elevation is obtained. Other flow quantities, such as pressure, flow velocity components, and bottom shear stress, are expressed in terms of the surface elevation. The steady-state version of the extended KdV equation has eigensolutions that describe stationary solitary waves. Time-dependent solutions of the extended KdV equation provide a means for discriminating between stable and unstable stationary solitary waves. Solutions of initial value problems show that there are transient solutions that approach asymptotically the stable stationary solitary wave, whereas other transient solutions decay asymptotically with increasing time.

The present analysis extends the author’s earlier work (Lappa, Phys. Fluids, vol. 25, 2003, 012101; Lappa, Chaos, vol. 23, 2003, 013105) on the properties of patterns formed by the spontaneous accumulation and ordering of solid particles in certain types of flow (with a toroidal structure and a travelling wave propagating in the azimuthal direction) by considering the potential impact of ‘vibrations’ (g-jitters) on such dynamics. It is shown that a kaleidoscope of possible variants exist whose nature and variety calls for a concerted analysis using the tools of computational fluid dynamics in synergy with dimensional arguments and existing theories on the effect of periodic accelerations on fluid systems. A possible categorization of the observed phenomena is introduced according to the type and scale of ‘defects’ displayed by the emerging particle aggregates with respect to unperturbed (vibration-less) conditions. It is shown that the resulting degree of ‘turbulence’ depends essentially on the direction $(\phi )$, amplitude $(\gamma )$ and frequency $(\varpi )$ of the applied inertial disturbance. A range of amplitudes and frequencies exist where the formation of recognizable particle structures is prevented. A quantitative map (in the $\gamma \text{{\ndash}} \varpi $ plane) for their occurrence is derived with the express intent of supporting the optimization of future experiments to be performed in space.

Direct numerical simulations (DNS) of turbulent boundary layers over isothermally heated walls were performed, and the effect of viscosity stratification on the turbulence statistics and skin friction were investigated. An empirical relation for temperature-dependent viscosity for water was adopted. Based on the free-stream temperature (30°C), two wall temperatures (70°C and 99°C) were selected. In the heated flows, the turbulence energy diminishes in the buffer layer, but increases near the wall. The reduction in turbulence kinetic energy in the buffer layer is accompanied by smaller levels of Reynolds shear stresses and, hence, weaker turbulence production. The enhanced turbulence energy near the wall is attributed to enhanced transfer of energy via additional diffusion-like terms due to the viscosity stratification. Despite the lower fluid viscosity near the wall, dissipation is also increased owing to the augmented near-wall fine-scale motion. Wall heating results in reduction in the skin-friction coefficient by up to 26 %. An evaluation of the different contributions to the skin friction demonstrates that drag reduction is primarily due to the changes in the Reynolds shear stresses across the boundary layer. Quadrant and octant analyses showed that ejections (Q2) and sweeps (Q4) are significantly reduced, a result further supported by an examination of outer vortical structures from linear stochastic estimation of the ejection events and spanwise vortices.

Non-hydrostatic multi-layer models have become a popular tool in describing wave transformation from deep water to the surf zone, but the numerical approach lacks a theoretical framework to guide implementation and assist interpretation of the results. In this paper, we formulate a non-hydrostatic model in an analytical form for the derivation and examination of dispersive and nonlinear properties. Depth integration of the dimensionless continuity and Euler equations over each layer yields the conventional multi-layer formulation. A variable transformation converts the conventional form into an integrated series form, which provides separate descriptions of flux- and dispersion-dominated processes. Substitution of the non-hydrostatic pressure and vertical velocity in the governing equations by high-order derivatives of the horizontal velocity and surface elevation provides a direct comparison with the Boussinesq equations published in the literature. Implementation of a perturbation expansion extracts the first- and second-order governing equations with respect to the nonlinear parameter. Based on that, we derive analytical solutions of the linear dispersion and the second-order super- and sub-harmonics for up to three layers and optimize the solutions in terms of the layer arrangement. In relation to the Boussinesq equations at comparable orders of expansion, the two- and three-layer models provide slightly higher errors in shallow and intermediate water in terms of dispersion and super-harmonics, but show superior performance in describing sub-harmonics in deep water.

By studying the dynamics of the streamwise pressure gradient at the wavefront of travelling interfacial waves, we investigate the formation and evolution of backflow regions for the sinusoidal and teardrop-shaped surface wave regimes of laminar falling liquid films. The magnitude of the wavefront streamwise pressure gradient grows as the flow inlet disturbance increases in amplitude and steepness. At large enough values, the adverse pressure gradient induces flow separation and subsequently backflow at the large-amplitude wavefront. The backflow region evolves from a closed circulation to an open vortex as the wave grows to saturation. The dynamics of the streamwise pressure gradient at the sinusoidal wavefront approaches a stable fixed point at saturation. Thus, the open vortex retains its structure as the wave continues downstream. The streamwise pressure gradient at the wavefront of the teardrop-shaped pulse evolves similarly to a time-periodic function with multiple minima/maxima. This phenomenon is a consequence of the interaction between the teardrop-shaped wave and newly formed preceding capillary waves. The nature of the teardrop pulse–capillary wave interaction is such that a decrease in magnitude of the streamwise pressure gradient at the teardrop-shaped wavefront is followed by an increase at the capillary wavefront and vice versa. The increased adverse pressure gradient at the capillary wavefront induces a second open vortex backflow, while the teardrop-shaped wavefront’s open vortex reverts to a closed circulation. This interaction between the waves continues as the teardrop pulse–capillary wavetrain travels downstream, leading to multiple capillary waves and backflow regions.

We use the boundary element method to study the low-Reynolds-number locomotion of a spherical model microorganism in a circular tube. The swimmer propels itself by tangential or normal surface motion in a tube whose radius is of the order of the swimmer size. Hydrodynamic interactions with the tube walls significantly affect the average swimming speed and power consumption of the model microorganism. In the case of swimming parallel to the tube axis, the locomotion speed is always reduced (respectively, increased) for swimmers with tangential (respectively, normal) deformation. In all cases, the rate of work necessary for swimming is increased by confinement. Swimmers with no force dipoles in the far field generally follow helical trajectories, solely induced by hydrodynamic interactions with the tube walls, and in qualitative agreement with recent experimental observations for Paramecium. Swimmers of the puller type always display stable locomotion at a location which depends on the strength of their force dipoles: swimmers with weak dipoles (small $\alpha $) swim in the centre of the tube while those with strong dipoles (large $\alpha $) swim near the walls. In contrast, pusher swimmers and those employing normal deformation are unstable and end up crashing into the walls of the tube. Similar dynamics is observed for swimming into a curved tube. These results could be relevant for the future design of artificial microswimmers in confined geometries.

Numerical simulations of the linear and nonlinear two-dimensional Navier–Stokes equations, and linear stability theory are used to parametrically investigate hypersonic boundary layers over ultrasonic absorptive coatings. The porous coatings consist of a uniform array of rectangular pores (slots) with a range of porosities and pore aspect ratios. For the numerical simulations, temporally (rather than spatially) evolving boundary layers are considered and we provide evidence that this approximation is appropriate for slowly growing second-mode instabilities. We consider coatings operating in the typical regime where the pores are relatively deep and acoustic waves and second-mode instabilities are attenuated by viscous effects inside the pores, as well as regimes with phase cancellation or reinforcement associated with reflection of acoustic waves from the bottom of the pores. These conditions are defined as attenuative and cancellation/reinforcement regimes, respectively. The focus of the present study is on the cases which have not been systematically studied in the past, namely the reinforcement regime (which represents a worst-case scenario, i.e. minimal second-mode damping) and the cancellation regime (which corresponds to the configuration with the most potential improvement). For all but one of the cases considered, the linear simulations show good agreement with the results of linear instability theory that employs an approximate porous-wall boundary condition, and confirm that the porous coating stabilizing performance is directly related to their acoustic scattering performance. A particular case with relatively shallow pores and very high porosity showed the existence of a shorter-wavelength instability that was not initially predicted by theory. Our analysis shows that this new mode is associated with acoustic resonances in the pores and can be more unstable than the second mode. Modifications to the theoretical model are suggested to account for the new mode and to provide estimates of the porous coating parameters that avoid this detrimental instability. Finally, nonlinear simulations confirm the conclusions of the linear analysis; in particular, we did not observe any tripping of the boundary layer by small-scale disturbances associated with individual pores.

We performed numerical simulations of dissolved oxygen (DO) transfer from a turbulent flow, driven by periodic boundary-layer turbulence in the intermittent regime, to underlying DO-absorbing organic sediment layers. A uniform initial distribution of oxygen is left to decay (with no re-aeration) as the turbulent transport supplies the sediment with oxygen from the outer layers to be absorbed. A very thin diffusive sublayer at the sediment–water interface (SWI), caused by the high Schmidt number of DO in water, limits the overall decay rate. A decomposition of the instantaneous decaying turbulent scalar field is proposed, which results in the development of similarity solutions that collapse the data in time. The decomposition is then tested against the governing equations, leading to a rigorous procedure for the extraction of an ergodic turbulent scalar field. The latter is composed of a statistically periodic and a steady non-decaying field. Temporal averaging is used in lieu of ensemble averaging to evaluate flow statistics, allowing the investigation of turbulent mixing dynamics from a single flow realization. In spite of the highly unsteady state of turbulence, the monotonically decaying component is surprisingly consistent with experimental and numerical correlations valid for steady high-Schmidt-number turbulent mass transfer. Linearly superimposed onto it is the statistically periodic component, which incorporates all the features of the non-equilibrium state of turbulence. It is modulated by the evolution of the turbulent coherent structures driven by the oscillating boundary layer in the intermittent regime, which are responsible for the violent turbulent production mechanisms. These cause, in turn, a rapid increase of the turbulent mass flux at the edge of the diffusive sublayer. This outer-layer forcing mechanism drives a periodic accumulation of high scalar concentration levels in the near-wall region. The resulting modulated scalar flux across the SWI is delayed by a quarter of a cycle with respect to the wall-shear stress, consistently with the non-equilibrium state of the turbulent mixing.

Nonlinear stratified spindown of an along-isobath current over an insulated slope is shown to develop asymmetries in the vertical circulation and vertical relative vorticity field. During spindown, cyclonic vorticity is weakened to a greater extent than anticyclonic vorticity near the boundary because of buoyancy advection. As a consequence, Ekman pumping is weakened over Ekman suction. Momentum advection can weaken Ekman pumping and strengthen Ekman suction. Time-dependent feedback between the geostrophic flow and the frictional secondary circulation induces asymmetry in cyclonic and anticyclonic vorticity away from the boundary. Buoyancy advection over a slope can modify the secondary circulation such that anticyclonic vorticity decays faster than cyclonic vorticity outside the boundary layer. In contrast, momentum advection can cause cyclonic vorticity to spin down faster than anticyclonic vorticity. A scaling and analytical solutions are derived for when buoyancy advection over a slope can have a more significant impact than momentum advection on these asymmetries. In order to test this scaling and analytical solutions, numerical experiments are run in which both buoyancy and momentum advection are active. These solutions are contrasted with homogeneous or stratified spindown over a flat bottom, in which momentum advection controls the asymmetries. These results are applied to ocean currents over continental shelves and slopes.

Consequences of density stratification are studied for an ideal (Euler) incompressible fluid, confined to move under gravity between rigid lids but otherwise free to move along horizontal directions. Initial conditions that generate horizontal pressure imbalances in a laterally unbounded domain are examined. The aim is to show analytically the existence of classes of initial data for which total horizontal momentum evolves in time, even though only vertical forces act on the fluid in this set-up. A simple class of such initial conditions, leading to momentum evolution, is identified by systematic asymptotic expansions of the governing inhomogeneous Euler equations in the small-density-variation limit. These results for Euler equations are compared and confirmed with long-wave asymptotic models, which can handle arbitrary density variations and provide closed-form mathematical expressions for limiting cases. In particular, the role of wave dispersion arising from the fluid inertia is captured by the long-wave models, even for short-time dynamics emanating from initial conditions outside the models’ asymptotic range of validity. These results are compared with direct numerical simulations for variable-density Euler fluids, which further validate the numerical algorithms and the analysis.

This paper is concerned with modelling the effects of swirling flow on turbomachinery noise. We develop an acoustic analogy to predict sound generation in a swirling and sheared base flow in an annular duct, including the presence of moving solid surfaces to account for blade rows. In so doing we have extended a number of classical earlier results, including Ffowcs Williams & Hawkings’ equation in a medium at rest with moving surfaces, and Lilley’s equation for a sheared but non-swirling jet. By rearranging the Navier–Stokes equations we find a single equation, in the form of a sixth-order differential operator acting on the fluctuating pressure field on the left-hand side and a series of volume and surface source terms on the right-hand side; the form of these source terms depends strongly on the presence of swirl and radial shear. The integral form of this equation is then derived, using the Green’s function tailored to the base flow in the (rigid) duct. As is often the case in duct acoustics, it is then convenient to move into temporal, axial and azimuthal Fourier space, where the Green’s function is computed numerically. This formulation can then be applied to a number of turbomachinery noise sources. For definiteness here we consider the noise produced downstream when a steady distortion flow is incident on the fan from upstream, and compare our results with those obtained using a simplistic but commonly used Doppler correction method. We show that in all but the simplest case the full inclusion of swirl within an acoustic analogy, as described in this paper, is required.

The influence of time-periodic vibrations on long Marangoni waves in two-layer films is investigated. The problem is governed by a system of nonlinear equations obtained in the framework of the lubrication approximation. Periodic boundary conditions are applied on the boundaries of the computational region. The development of instabilities is investigated by means of nonlinear simulations. Excitation of two-dimensional and three-dimensional subharmonic wavy regimes is studied. A new phenomenon, the excitation of nonlinear waves with a temporal period that is four times larger than that of the gravity modulation, is revealed.

A hybrid lattice-Boltzmann spectrin-link (LB–SL) method is used to simulate dense suspensions of red blood cells (RBCs) for investigating rheological properties of blood. RBC membranes are modelled using a coarse-grained SL method and are filled with a viscous Newtonian fluid solution with viscosity five times that of the suspending fluid. Relative viscosities, normal stress differences, and particle pressures are reported for a range of capillary numbers at a physiologically realistic haematocrit value of approximately 42.5 %. Viscosity shear thinning is demonstrated for shear rates ranging from 14 to 440  s−1 and is shown to be affected by the orientation and bending modulus of RBCs. The particle-phase pressure undergoes a change in sign from positive to negative as the shear rate is increased. The particle-phase normal stress tensor values show that there is a transition from compressive to tensile states in the flow direction as the shear rate is increased. The normal stress differences are notably different from those recently reported for deformable capsule suspensions using a similar methodology, which suggests that the bending stiffness and the biconcave shape of RBCs affect the rheology of blood.

We study the long-term evolution of weakly nonlinear random gravity water wave fields developing with and without wind forcing. The focus of the work is on deriving, from first principles, the evolution of the departure of the field statistics from Gaussianity. Higher-order statistical moments of elevation (skewness and kurtosis) are used as a measure of this departure. Non-Gaussianity of a weakly nonlinear random wave field has two components. The first is due to nonlinear wave–wave interactions. We refer to this component as ‘dynamic’, since it is linked to wave field evolution. The other component is due to bound harmonics. It is non-zero for every wave field with finite amplitude, contributes both to skewness and kurtosis of gravity water waves and can be determined entirely from the instantaneous spectrum of surface elevation. The key result of the work, supported both by direct numerical simulation (DNS) and by the analysis of simulated and experimental (JONSWAP) spectra, is that in generic situations of a broadband random wave field the dynamic contribution to kurtosis is small in absolute value, and negligibly small compared with the bound harmonics component. Therefore, the latter dominates, and both skewness and kurtosis can be obtained directly from the instantaneous wave spectra. Thus, the departure of evolving wave fields from Gaussianity can be obtained from evolving wave spectra, complementing the capability of forecasting spectra and capitalizing on the existing methodology. We find that both skewness and kurtosis are significant for typical oceanic waves; the non-zero positive kurtosis implies a tangible increase of freak wave probability. For random wave fields generated by steady or slowly varying wind and for swell the derived large-time asymptotics of skewness and kurtosis predict power law decay of the moments. The exponents of these laws are determined by the degree of homogeneity of the interaction coefficients. For all self-similar regimes the kurtosis decays twice as fast as the skewness. These formulae complement the known large-time asymptotics for spectral evolution prescribed by the Hasselmann equation. The results are verified by the DNS of random wave fields based on the Zakharov equation. The predicted asymptotic behaviour is shown to be very robust: it holds both for steady and gusty winds.